{"id":644,"date":"2024-04-23T19:57:14","date_gmt":"2024-04-23T19:57:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=644"},"modified":"2025-01-16T19:30:42","modified_gmt":"2025-01-16T19:30:42","slug":"exponents-and-scientific-notation-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponents-and-scientific-notation-fresh-take\/","title":{"raw":"Exponents and Scientific Notation: Fresh Take","rendered":"Exponents and Scientific Notation: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Understand and use the rules for exponents<\/li>\r\n \t<li>Change numbers between scientific notation and standard notation<\/li>\r\n \t<li>Solve calculations using scientific notation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Notation<\/h2>\r\nWe use exponential notation to write repeated multiplication. For example [latex]10\\cdot10\\cdot10[\/latex] can be written more succinctly as [latex]10^{3}[\/latex]. The 10 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>base<\/b>. The 3 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>exponent<\/b>. The expression [latex]10^{3}[\/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.\r\n<p style=\"text-align: center;\">[latex]\\text{base}\\rightarrow10^{3\\leftarrow\\text{exponent}}[\/latex]<\/p>\r\n[latex]10^{3}[\/latex] is read as \u201c[latex]10[\/latex] to the third power\u201d or \u201c[latex]10[\/latex] cubed.\u201d It means [latex]10\\cdot10\\cdot10[\/latex], or [latex]1,000[\/latex].\r\n\r\n[latex]8^{2}[\/latex]\u00a0is read as \u201c[latex]8[\/latex] to the second power\u201d or \u201c[latex]8[\/latex] squared.\u201d It means [latex]8\\cdot8[\/latex], or [latex]64[\/latex].\r\n\r\n[latex]5^{4}[\/latex]\u00a0is read as \u201c[latex]5[\/latex] to the fourth power.\u201d It means [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex].\r\n\r\n[latex]b^{5}[\/latex]\u00a0is read as \u201c[latex]b[\/latex] to the fifth power.\u201d It means [latex]{b}\\cdot{b}\\cdot{b}\\cdot{b}\\cdot{b}[\/latex]. Its value will depend on the value of [latex]b[\/latex].\r\n\r\nThe exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[\/latex],\u00a0only the[latex]y[\/latex] is affected by the [latex]4[\/latex]. [latex]xy^{4}[\/latex]\u00a0means [latex]{x}\\cdot{y}\\cdot{y}\\cdot{y}\\cdot{y}[\/latex]. The [latex]x[\/latex] in this term is a <strong>coefficient<\/strong> of [latex]y[\/latex].\r\n\r\nIf the exponential expression is negative, such as [latex]\u22123^{4}[\/latex], it means [latex]\u2013\\left(3\\cdot3\\cdot3\\cdot3\\right)[\/latex] or [latex]\u221281[\/latex].\r\n\r\nIf [latex]\u22123[\/latex] is to be the base, it must be written as [latex]\\left(\u22123\\right)^{4}[\/latex], which means [latex]\u22123\\cdot\u22123\\cdot\u22123\\cdot\u22123[\/latex], or [latex]81[\/latex].\r\n\r\nLikewise,\u00a0[latex]\\left(\u2212x\\right)^{4}=\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)=x^{4}[\/latex], while [latex]\u2212x^{4}=\u2013\\left(x\\cdot x\\cdot x\\cdot x\\right)[\/latex].\r\n\r\nYou can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.\r\n\r\n<section class=\"textbox example\">Simplify:\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]{5}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{9}^{1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"153466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"153466\"]\r\n<ol>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{5}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]3[\/latex] factors of [latex]5[\/latex].<\/td>\r\n<td>[latex]5\\cdot 5\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]125[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{9}^{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]1[\/latex] factor of [latex]9[\/latex].<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Simplify:\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]{\\left({\\Large\\frac{7}{8}}\\right)}^{2}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(0.74\\right)}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"153461\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"153461\"]\r\n<ol>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 15.2334px;\">\r\n<td style=\"height: 15.2334px;\"><\/td>\r\n<td style=\"height: 15.2334px;\">[latex]{\\left({\\Large\\frac{7}{8}}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Multiply two factors.<\/td>\r\n<td style=\"height: 15px;\">[latex]\\left({\\Large\\frac{7}{8}}\\right)\\left({\\Large\\frac{7}{8}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]{\\Large\\frac{49}{64}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(0.74\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply two factors.<\/td>\r\n<td>[latex]\\left(0.74\\right)\\left(0.74\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]0.5476[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Simplify:\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/li>\r\n \t<li>[latex]{-3}^{4}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"152453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152453\"]\r\n<ol>\r\n \t<li>&lt;table \"&gt;\r\n[latex]{\\left(-3\\right)}^{4}[\/latex]Multiply four factors of [latex]\u22123[\/latex].[latex]\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)[\/latex]Simplify.[latex]81[\/latex]<\/li>\r\n \t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{-3}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply two factors.<\/td>\r\n<td>[latex]-\\left(3\\cdot 3\\cdot 3\\cdot 3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\nNotice the similarities and differences in parts 1 and 2. Why are the answers different? In part 1 the parentheses tell us to raise the [latex](\u22123)[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power. In part 2 we raise only the [latex]3[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power and then find the opposite.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Evaluate [latex]x^{3}[\/latex] if [latex]x=\u22124[\/latex].[reveal-answer q=\"86290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"86290\"]Substitute [latex]\u22124[\/latex] for the variable [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(\u22124\\right)^{3}[\/latex]<\/p>\r\nEvaluate. Note how placing parentheses around the [latex]\u22124[\/latex] means the negative sign also gets multiplied.\r\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124=\u221264[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x^{3}=\u221264[\/latex] when [latex]x=\u22124[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>The Product Rule for Exponents<\/h2>\r\nWhat happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents, [latex]3[\/latex] and [latex]4[\/latex].\r\n\r\nWhat about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, [latex]8[\/latex] is the sum of the original two exponents. This concept can be generalized in the following way: For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex],\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhhdhdgf-hA9AT7QsXWo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/hA9AT7QsXWo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhhdhdgf-hA9AT7QsXWo\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843107&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hhhdhdgf-hA9AT7QsXWo&vembed=0&video_id=hA9AT7QsXWo&video_target=tpm-plugin-hhhdhdgf-hA9AT7QsXWo'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Exponential+Expressions+Using+the+Product+Property+of+Exponents_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Exponential Expressions Using the Product Property of Exponents\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>The Quotient (Division) Rule for Exponents<\/h2>\r\nLet\u2019s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\r\nYou can rewrite the expression as: [latex] \\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of [latex]4[\/latex] in the numerator and denominator: [latex] \\displaystyle [\/latex]\r\n\r\nFinally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, [latex]3[\/latex], is the difference between the two exponents in the original expression, [latex]5[\/latex] and [latex]2[\/latex].\r\n\r\nSo,\u00a0[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].\r\n\r\nBe careful that you subtract the exponent in the denominator from the exponent in the numerator.\r\n\r\nSo, to divide two exponential terms with the same base, subtract the exponents.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcaehbea-Jmf-CPhm3XM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Jmf-CPhm3XM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcaehbea-Jmf-CPhm3XM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843108&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gcaehbea-Jmf-CPhm3XM&vembed=0&video_id=Jmf-CPhm3XM&video_target=tpm-plugin-gcaehbea-Jmf-CPhm3XM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Exponential+Expressions+Using+the+Quotient+Property+of+Exponents_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Exponential Expressions Using the Quotient Property of Exponents\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Raise Powers to Powers<\/h2>\r\nLet\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is [latex]4[\/latex], so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).\r\n\r\n[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of [latex]5[\/latex] to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].\r\n\r\nSo, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).\r\n\r\nLikewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]\r\n\r\nThis leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].\r\n\r\n<section class=\"textbox example\">Simplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].[reveal-answer q=\"841688\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841688\"]Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.\r\n<p style=\"text-align: center;\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>Answer: [latex]6\\left(c^{4}\\right)^{2}=6c^{8}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Raise a Product to a Power<\/h2>\r\nSimplify this expression.\r\n<p style=\"text-align: center;\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\r\nNotice that the exponent is applied to each factor of [latex]2a[\/latex]. So, we can eliminate the middle steps.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\r\nThe product of two or more numbers raised to a power is equal to the product of each number raised to the same power.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdhfbfeg-Hgu9HKDHTUA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Hgu9HKDHTUA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fdhfbfeg-Hgu9HKDHTUA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843109&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fdhfbfeg-Hgu9HKDHTUA&vembed=0&video_id=Hgu9HKDHTUA&video_target=tpm-plugin-fdhfbfeg-Hgu9HKDHTUA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Exponential+Expressions+Using+the+Power+Property+of+Exponents_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Exponential Expressions Using the Power Property of Exponents\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Raise a Quotient to a Power<\/h2>\r\nNow let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex] \\displaystyle \\frac{3}{4}[\/latex] and raise it to the [latex]3[\/latex]<sup>rd<\/sup> power.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\r\nYou can see that raising the quotient to the power of [latex]3[\/latex] can also be written as the numerator ([latex]3[\/latex]) to the power of [latex]3[\/latex], and the denominator ([latex]4[\/latex]) to the power of [latex]3[\/latex].\r\n\r\nSimilarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)=\\frac{a\\cdot a\\cdot a\\cdot a}{b\\cdot b\\cdot b\\cdot b}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\r\nWhen a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bdgghedd-ZbxgDRV35dE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZbxgDRV35dE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bdgghedd-ZbxgDRV35dE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843110&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bdgghedd-ZbxgDRV35dE&vembed=0&video_id=ZbxgDRV35dE&video_target=tpm-plugin-bdgghedd-ZbxgDRV35dE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Fractions+Raised+to+Powers+(Positive+Exponents+Only)+Version+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>The Zero Exponent Rule<\/h2>\r\n<h3>What if the exponent is zero?<\/h3>\r\nTo see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of [latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\r\nIf we were to simplify the original expression using the quotient rule, we would have\r\n<p style=\"text-align: center;\">[latex]\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/p>\r\nIf we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.\r\n<p style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/p>\r\nThe sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or [latex]DNE[\/latex] (Does Not Exist).\r\n\r\n<section class=\"textbox example\">Evaluate [latex]2x^{0}[\/latex] if [latex]x=9[\/latex][reveal-answer q=\"324798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324798\"]Substitute [latex]9[\/latex] for the variable [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]2\\cdot9^{0}[\/latex]<\/p>\r\nEvaluate [latex]9^{0}[\/latex]. Multiply.\r\n<p style=\"text-align: center;\">[latex]2\\cdot1=2[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>Answer: [latex]2x^{0}=2[\/latex], if [latex]x=9[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbhdfeeb-jKihp_DVDa0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/jKihp_DVDa0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbhdfeeb-jKihp_DVDa0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843126&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fbhdfeeb-jKihp_DVDa0&vembed=0&video_id=jKihp_DVDa0&video_target=tpm-plugin-fbhdfeeb-jKihp_DVDa0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Evaluate+and+Simplify+Expressions+Using+the+Zero+Exponent+Rule_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate and Simplify Expressions Using the Zero Exponent Rule\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>The Negative Exponent Rule<\/h2>\r\nGiven a quotient like\u00a0[latex] \\displaystyle \\frac{{{2}^{m}}}{{{2}^{n}}}[\/latex] what happens when [latex]n[\/latex] is larger than [latex]m[\/latex]? We will need to use the <em>negative rule of exponents<\/em> to simplify the expression so that it is easier to understand.\r\n\r\nLet's look at an example to clarify this idea. Given the expression:\r\n<p style=\"text-align: center;\">[latex]\\frac{{h}^{3}}{{h}^{5}}[\/latex]<\/p>\r\nExpand the numerator and denominator, all the terms in the numerator will cancel to [latex]1[\/latex], leaving two [latex]h[\/latex]s multiplied in the denominator, and a numerator of [latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\frac{{h}^{3}}{{h}^{5}}\\,\\,\\,=\\,\\,\\,\\frac{h\\cdot{h}\\cdot{h}}{h\\cdot{h}\\cdot{h}\\cdot{h}\\cdot{h}} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}}{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}\\cdot {h}\\cdot {h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{h\\cdot{h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{{h}^{2}} \\end{array}[\/latex]<\/p>\r\nWe could have also applied the quotient rule from the last section, to obtain the following result:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{h^{3}}{h^{5}}\\,\\,\\,=\\,\\,\\,h^{3-5}\\\\\\\\=\\,\\,\\,h^{-2}\\,\\,\\end{array}[\/latex]<\/p>\r\nPutting the answers together, we have [latex]{h}^{-2}=\\frac{1}{{h}^{2}}[\/latex]. This is true when [latex]h[\/latex], or any variable, is a real number and is not zero.\r\n\r\n<section class=\"textbox example\">Write [latex]\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}[\/latex] with positive exponents.\r\n<p style=\"text-align: left;\">[reveal-answer q=\"219981\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"219981\"]<\/p>\r\n<p style=\"text-align: left;\">Use the quotient rule to subtract the exponents of terms with like bases.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}={t}^{3-8}\\\\={t}^{-5}\\,\\,\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Write the expression with positive exponents by putting the term with the negative exponent in the denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]=\\frac{1}{{t}^{5}}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>Answer: [latex]\\frac{1}{{t}^{5}}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">Simplify [latex]{\\left(\\frac{1}{3}\\right)}^{-2}[\/latex].[reveal-answer q=\"998337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"998337\"]Apply the power property of exponents.\r\n<p style=\"text-align: center;\">[latex]\\frac{{1}^{-2}}{{3}^{-2}}[\/latex]<\/p>\r\nWrite each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.\r\n<p style=\"text-align: center;\">[latex]\\frac{{3}^{2}}{{1}^{2}}{ = }\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}{ = }\\frac{9}{1}{ = }{9}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<center>Answer: [latex]9[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efaagccc-WvFlHjlIITg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/WvFlHjlIITg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-efaagccc-WvFlHjlIITg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843127&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-efaagccc-WvFlHjlIITg&vembed=0&video_id=WvFlHjlIITg&video_target=tpm-plugin-efaagccc-WvFlHjlIITg'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Negative+Exponents+-+Basics_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Negative Exponents - Basics\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Convert Standard Notation to Scientific Notation<\/h2>\r\nRemember working with place value for whole numbers and decimals? Our number system is based on powers of [latex]10[\/latex]. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens\u2014tenths, hundredths, thousandths, and so on.\r\n\r\nConsider the numbers [latex]4000[\/latex] and [latex]0.004[\/latex]. We know that [latex]4000[\/latex] means [latex]4\u00d71000[\/latex] and [latex]0.004[\/latex] means [latex]4 \u00d7 \\frac{1}{1000}[\/latex]. If we write the [latex]1000[\/latex] as a power of ten in exponential form, we can rewrite these numbers in this way:\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{cc} \\hfill 4000 &amp;&amp;&amp;&amp;&amp;&amp; 0.004 \\hfill \\\\ 4 \\text{ x } 1000 &amp;&amp;&amp;&amp;&amp;&amp; 4 \\text{ x } \\frac{1}{1000} \\hfill \\\\ 4 \\text{ x } 10^3 &amp;&amp;&amp;&amp;&amp;&amp; 4 \\text{ x } \\frac{1}{10^3} \\hfill \\\\ &amp;&amp;&amp;&amp;&amp;&amp; 4 \\text{ x } 10^{-3} \\hfill \\\\ \\end{array}[\/latex]<\/p>\r\nWhen a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than [latex]10[\/latex], and the second factor is a power of [latex]10[\/latex] written in exponential form, it is said to be in scientific notation.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>a general note: scientific notation<\/h3>\r\nA number is written in <strong>scientific notation<\/strong> if it is written in the form [latex]a\\times {10}^{n}[\/latex], where [latex]1\\le |a|&lt;10[\/latex] and [latex]n[\/latex] is an integer.\r\n\r\n<\/div>\r\n<\/section>Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier. If we look at what happened to the decimal point, we can see a method to easily convert from standard notation to scientific notation.\r\n\r\n<center><img class=\"alignnone size-full wp-image-166\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5838\/2022\/11\/28165215\/QR-Sci-Not-Dig-Deeper-1.jpg\" alt=\"On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, \u201cMoved the decimal point 3 places to the left.\u201d On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, \u201cMoved the decimal point 3 places to the right.\" width=\"449\" height=\"120\" \/><\/center>&nbsp;\r\n\r\nIn both cases, the decimal was moved [latex]3[\/latex] places to get the first factor, [latex]4[\/latex], by itself.\r\n\r\nThe power of [latex]10 [\/latex] is positive when the number is larger than [latex]1[\/latex]: [latex]4000=4\u00d710^3[\/latex].\r\nThe power of [latex]10 [\/latex] is negative when the number is between [latex]0[\/latex] and [latex]1[\/latex]: [latex]0.004=4\u00d710^{-3}[\/latex].\r\n\r\nWatch the following video to see more examples of writing numbers in scientific notation.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-geacbdhc-fsNu3AdIgdk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/fsNu3AdIgdk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-geacbdhc-fsNu3AdIgdk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843128&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-geacbdhc-fsNu3AdIgdk&vembed=0&video_id=fsNu3AdIgdk&video_target=tpm-plugin-geacbdhc-fsNu3AdIgdk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Examples+-+Write+a+Number+in+Scientific+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Write a Number in Scientific Notation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox seeExample\">Write each number in scientific notation.\r\n<ol>\r\n \t<li>Distance to Andromeda Galaxy from Earth: [latex]24,000,000,000,000,000,000,000[\/latex] m<\/li>\r\n \t<li>Diameter of Andromeda Galaxy: [latex]1,300,000,000,000,000,000,000[\/latex] m<\/li>\r\n \t<li>Number of stars in Andromeda Galaxy: [latex]1,000,000,000,000[\/latex]<\/li>\r\n \t<li>Diameter of electron: [latex]0.00000000000094[\/latex] m<\/li>\r\n \t<li>Probability of being struck by lightning in any single year: [latex]0.00000143[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"500341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500341\"]\r\n\r\nObserve that, if the given number is greater than [latex]1[\/latex], as in examples a\u2013c, the exponent of [latex]10[\/latex] is positive; and if the number is less than [latex]1[\/latex], as in examples d\u2013e, the exponent is negative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2187[\/ohm2_question]<\/section>\r\n<h2>Convert Scientific Notation to Standard Notation<\/h2>\r\nHow can we convert from scientific notation to standard notation? Let\u2019s look at two numbers written in scientific notation and see.\r\n\r\n<center>[latex] \\begin{array}{cc} \\hfill 9.12 \\text{ x } 10^{4} &amp;&amp;&amp;&amp;&amp;&amp; 9.12 \\text{ x } 10^{-4} \\hfill \\\\ 9.12 \\text{ x } 10,000 &amp;&amp;&amp;&amp;&amp;&amp; 9.12 \\text{ x } 0.0001 \\hfill \\\\ 91,200 &amp;&amp;&amp;&amp;&amp;&amp; 0.000912 \\hfill \\\\ \\end{array}[\/latex]<\/center><section class=\"textbox questionHelp\"><strong>How To: Convert Scientific Notation to Standard Notation<\/strong>\r\n<ul>\r\n \t<li>Step 1: Determine the exponent, [latex]n[\/latex], on the factor [latex]10[\/latex].<\/li>\r\n \t<li>Step 2: Move the decimal [latex]n[\/latex] places, adding zeros if needed.<\/li>\r\n \t<li>Step 3: Check your answer.<\/li>\r\n<\/ul>\r\n<\/section>If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form \u2013 also known as standard notation.\r\n\r\n<center><img class=\"alignnone size-full wp-image-192\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5838\/2022\/11\/28175455\/5eea11b73e531ec9788f076d12a39212b3192193.jpg\" alt=\"On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.\" width=\"498\" height=\"65\" \/><\/center>&nbsp;\r\n\r\nIn both cases, the decimal point moved [latex]4[\/latex] places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.\r\n\r\nWatch the following video to see more examples of writing scientific notation in standard notation.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aefgdhhd-8BX0oKUMIjw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/8BX0oKUMIjw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aefgdhhd-8BX0oKUMIjw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12539606&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aefgdhhd-8BX0oKUMIjw&vembed=0&video_id=8BX0oKUMIjw&video_target=tpm-plugin-aefgdhhd-8BX0oKUMIjw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Examples+-+Writing+a+Number+in+Decimal+Notation+When+Given+in+Scientific+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Writing a Number in Decimal Notation When Given in Scientific Notation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox seeExample\">Convert each number in scientific notation to standard notation.\r\n<ol>\r\n \t<li>[latex]3.547\\times {10}^{14}[\/latex]<\/li>\r\n \t<li>[latex]-2\\times {10}^{6}[\/latex]<\/li>\r\n \t<li>[latex]7.91\\times {10}^{-7}[\/latex]<\/li>\r\n \t<li>[latex]-8.05\\times {10}^{-12}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"500342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500342\"]\r\n1.\r\n[latex]\\begin{align}&amp;3.547\\times {10}^{14} \\\\ &amp;\\underset{\\to 14\\text{ places}}{{3.54700000000000}} \\\\ &amp;354,700,000,000,000 \\\\ \\text{ }\\end{align}[\/latex]\r\n\r\n2.\r\n[latex]\\begin{align}&amp;-2\\times {10}^{6} \\\\ &amp;\\underset{\\to 6\\text{ places}}{{-2.000000}} \\\\ &amp;-2,000,000 \\\\ \\text{ }\\end{align}[\/latex]\r\n\r\n3.\r\n[latex]\\begin{align}&amp;7.91\\times {10}^{-7} \\\\ &amp;\\underset{\\to 7\\text{ places}}{{0000007.91}} \\\\ &amp;0.000000791 \\\\ \\text{ }\\end{align}[\/latex]\r\n\r\n4.\r\n[latex]\\begin{align}&amp;-8.05\\times {10}^{-12} \\\\ &amp;\\underset{\\to 12\\text{ places}}{{-000000000008.05}} \\\\ &amp;-0.00000000000805 \\\\ \\text{ }\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2189[\/ohm2_question]<\/section>We use the Properties of Exponents to multiply and divide numbers in scientific notation.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>multiplying and diving numbers in scientific notation<\/h3>\r\nTo multiply numbers in scientific notation, we need to multiply the coefficients and add the powers of [latex]10[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo divide numbers in scientific notation, we need to divide the coefficients and subtract the powers of [latex]10[\/latex].\r\n\r\n<\/div>\r\n<\/section>Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in [latex]1[\/latex] L of water. Each water molecule contains [latex]3[\/latex] atoms ([latex]2[\/latex] hydrogen and [latex]1[\/latex] oxygen). The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and [latex]1[\/latex] L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops. Therefore, there are approximately [latex]3\\cdot \\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in [latex]1[\/latex] L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!\r\n\r\n<section class=\"textbox seeExample\">Perform the operations and write the answer in scientific notation.\r\n<ol>\r\n \t<li>[latex]\\left(8.14\\times {10}^{-7}\\right)\\left(6.5\\times {10}^{10}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"500343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500343\"]\r\n<ol>\r\n \t<li>[latex]\\begin{align}\\left(8.14 \\times 10^{-7}\\right)\\left(6.5 \\times 10^{10}\\right) &amp; =\\left(8.14 \\times 6.5\\right)\\left(10^{-7} \\times 10^{10}\\right) &amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; =\\left(52.91\\right)\\left(10^{3}\\right) &amp;&amp; \\text{Product rule of exponents} \\\\ &amp; =5.291 \\times 10^{4} &amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)&amp; = \\left(\\frac{4}{-1.52}\\right)\\left(\\frac{{10}^{5}}{{10}^{9}}\\right)&amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; = \\left(-2.63\\right)\\left({10}^{-4}\\right)&amp;&amp; \\text{Quotient rule of exponents} \\\\ &amp; = -2.63\\times {10}^{-4}&amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)&amp; = \\left(2.7\\times 6.04\\right)\\left({10}^{5}\\times {10}^{13}\\right)&amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; = \\left(16.308\\right)\\left({10}^{18}\\right)&amp;&amp; \\text{Product rule of exponents} \\\\ &amp; = 1.6308\\times {10}^{19}&amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)&amp; = \\left(\\frac{1.2}{9.6}\\right)\\left(\\frac{{10}^{8}}{{10}^{5}}\\right)&amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; = \\left(0.125\\right)\\left({10}^{3}\\right)&amp;&amp; \\text{Quotient rule of exponents} \\\\ &amp; = 1.25\\times {10}^{2}&amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)&amp; = \\left[3.33\\times \\left(-1.05\\right)\\times 5.62\\right]\\left({10}^{4}\\times {10}^{7}\\times {10}^{5}\\right) \\\\ &amp; \\approx \\left(-19.65\\right)\\left({10}^{16}\\right) \\\\ &amp; = -1.965\\times {10}^{17} \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video to see more examples of multiplying and dividing numbers in scientific notation.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beegacfg-yX6Mq9whsX0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/yX6Mq9whsX0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-beegacfg-yX6Mq9whsX0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=10294935&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-beegacfg-yX6Mq9whsX0&vembed=0&video_id=yX6Mq9whsX0&video_target=tpm-plugin-beegacfg-yX6Mq9whsX0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Scientific+Notation+-+Multiplication+and+Division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cScientific Notation - Multiplication and Division\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2190[\/ohm2_question]<\/section><section class=\"textbox seeExample\">An average human body contains around [latex]30,000,000,000,000[\/latex] red blood cells. Each cell measures approximately [latex]0.000008[\/latex] m long. Find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.\r\n[reveal-answer q=\"500344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500344\"]First, we must write both the number of red blood cells and the length in scientific notation.Number of red blood cells:[latex]30,000,000,000,000[\/latex] = [latex]3\\times {10}^{13}[\/latex]<em>Note: In order to get [latex]3[\/latex] by itself we moved the decimal place [latex]13[\/latex] times to the left making our exponent positive [latex]13[\/latex].<\/em>Length of red blood cells:[latex]0.000008 = 8\\times {10}^{-6}[\/latex]\r\n<em>Note: In order to get [latex]8[\/latex] by itself we moved the decimal place [latex]6[\/latex] times to the right making our exponent negative [latex]6[\/latex].<\/em> Now that we have both numbers in scientific notation, we can find the total length of all the cells laid end-to-end.To find the total length we must multiple the number of red blood cells by the length of each blood cell.[latex]\\begin{align}\\left(3 \\times 10^{13}\\right)\\left(8 \\times 10^{-6}\\right) &amp; =\\left(3 \\times 8\\right)\\left(10^{13} \\times 10^{-6}\\right) &amp;&amp; \\text{Commutative and associative properties of multiplication} \\\\ &amp; =\\left(24\\right)\\left(10^{7}\\right) &amp;&amp; \\text{Product rule of exponents} \\\\ &amp; =2.4 \\times 10^{8} &amp;&amp; \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]Number of cells: [latex]2.4\\times {10}^{8}[\/latex] m or [latex]240,000,000[\/latex] m.[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1842[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand and use the rules for exponents<\/li>\n<li>Change numbers between scientific notation and standard notation<\/li>\n<li>Solve calculations using scientific notation<\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Notation<\/h2>\n<p>We use exponential notation to write repeated multiplication. For example [latex]10\\cdot10\\cdot10[\/latex] can be written more succinctly as [latex]10^{3}[\/latex]. The 10 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>base<\/b>. The 3 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>exponent<\/b>. The expression [latex]10^{3}[\/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{base}\\rightarrow10^{3\\leftarrow\\text{exponent}}[\/latex]<\/p>\n<p>[latex]10^{3}[\/latex] is read as \u201c[latex]10[\/latex] to the third power\u201d or \u201c[latex]10[\/latex] cubed.\u201d It means [latex]10\\cdot10\\cdot10[\/latex], or [latex]1,000[\/latex].<\/p>\n<p>[latex]8^{2}[\/latex]\u00a0is read as \u201c[latex]8[\/latex] to the second power\u201d or \u201c[latex]8[\/latex] squared.\u201d It means [latex]8\\cdot8[\/latex], or [latex]64[\/latex].<\/p>\n<p>[latex]5^{4}[\/latex]\u00a0is read as \u201c[latex]5[\/latex] to the fourth power.\u201d It means [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex].<\/p>\n<p>[latex]b^{5}[\/latex]\u00a0is read as \u201c[latex]b[\/latex] to the fifth power.\u201d It means [latex]{b}\\cdot{b}\\cdot{b}\\cdot{b}\\cdot{b}[\/latex]. Its value will depend on the value of [latex]b[\/latex].<\/p>\n<p>The exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[\/latex],\u00a0only the[latex]y[\/latex] is affected by the [latex]4[\/latex]. [latex]xy^{4}[\/latex]\u00a0means [latex]{x}\\cdot{y}\\cdot{y}\\cdot{y}\\cdot{y}[\/latex]. The [latex]x[\/latex] in this term is a <strong>coefficient<\/strong> of [latex]y[\/latex].<\/p>\n<p>If the exponential expression is negative, such as [latex]\u22123^{4}[\/latex], it means [latex]\u2013\\left(3\\cdot3\\cdot3\\cdot3\\right)[\/latex] or [latex]\u221281[\/latex].<\/p>\n<p>If [latex]\u22123[\/latex] is to be the base, it must be written as [latex]\\left(\u22123\\right)^{4}[\/latex], which means [latex]\u22123\\cdot\u22123\\cdot\u22123\\cdot\u22123[\/latex], or [latex]81[\/latex].<\/p>\n<p>Likewise,\u00a0[latex]\\left(\u2212x\\right)^{4}=\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)=x^{4}[\/latex], while [latex]\u2212x^{4}=\u2013\\left(x\\cdot x\\cdot x\\cdot x\\right)[\/latex].<\/p>\n<p>You can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.<\/p>\n<section class=\"textbox example\">Simplify:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]{5}^{3}[\/latex]<\/li>\n<li>[latex]{9}^{1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q153466\">Show Solution<\/button><\/p>\n<div id=\"q153466\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{5}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]3[\/latex] factors of [latex]5[\/latex].<\/td>\n<td>[latex]5\\cdot 5\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]125[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{9}^{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]1[\/latex] factor of [latex]9[\/latex].<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Simplify:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]{\\left({\\Large\\frac{7}{8}}\\right)}^{2}[\/latex]<\/li>\n<li>[latex]{\\left(0.74\\right)}^{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q153461\">Show Solution<\/button><\/p>\n<div id=\"q153461\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr style=\"height: 15.2334px;\">\n<td style=\"height: 15.2334px;\"><\/td>\n<td style=\"height: 15.2334px;\">[latex]{\\left({\\Large\\frac{7}{8}}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Multiply two factors.<\/td>\n<td style=\"height: 15px;\">[latex]\\left({\\Large\\frac{7}{8}}\\right)\\left({\\Large\\frac{7}{8}}\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]{\\Large\\frac{49}{64}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(0.74\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply two factors.<\/td>\n<td>[latex]\\left(0.74\\right)\\left(0.74\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]0.5476[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Simplify:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/li>\n<li>[latex]{-3}^{4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q152453\">Show Solution<\/button><\/p>\n<div id=\"q152453\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>&lt;table &#8220;&gt;<br \/>\n[latex]{\\left(-3\\right)}^{4}[\/latex]Multiply four factors of [latex]\u22123[\/latex].[latex]\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)[\/latex]Simplify.[latex]81[\/latex]<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{-3}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply two factors.<\/td>\n<td>[latex]-\\left(3\\cdot 3\\cdot 3\\cdot 3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p>Notice the similarities and differences in parts 1 and 2. Why are the answers different? In part 1 the parentheses tell us to raise the [latex](\u22123)[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power. In part 2 we raise only the [latex]3[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power and then find the opposite.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Evaluate [latex]x^{3}[\/latex] if [latex]x=\u22124[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q86290\">Show Solution<\/button><\/p>\n<div id=\"q86290\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]\u22124[\/latex] for the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\u22124\\right)^{3}[\/latex]<\/p>\n<p>Evaluate. Note how placing parentheses around the [latex]\u22124[\/latex] means the negative sign also gets multiplied.<\/p>\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124=\u221264[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x^{3}=\u221264[\/latex] when [latex]x=\u22124[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>The Product Rule for Exponents<\/h2>\n<p>What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents, [latex]3[\/latex] and [latex]4[\/latex].<\/p>\n<p>What about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, [latex]8[\/latex] is the sum of the original two exponents. This concept can be generalized in the following way: For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex],\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhhdhdgf-hA9AT7QsXWo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/hA9AT7QsXWo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhhdhdgf-hA9AT7QsXWo\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843107&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hhhdhdgf-hA9AT7QsXWo&#38;vembed=0&#38;video_id=hA9AT7QsXWo&#38;video_target=tpm-plugin-hhhdhdgf-hA9AT7QsXWo\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Exponential+Expressions+Using+the+Product+Property+of+Exponents_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Exponential Expressions Using the Product Property of Exponents\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Quotient (Division) Rule for Exponents<\/h2>\n<p>Let\u2019s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\n<p>You can rewrite the expression as: [latex]\\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of [latex]4[\/latex] in the numerator and denominator: [latex]\\displaystyle[\/latex]<\/p>\n<p>Finally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, [latex]3[\/latex], is the difference between the two exponents in the original expression, [latex]5[\/latex] and [latex]2[\/latex].<\/p>\n<p>So,\u00a0[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].<\/p>\n<p>Be careful that you subtract the exponent in the denominator from the exponent in the numerator.<\/p>\n<p>So, to divide two exponential terms with the same base, subtract the exponents.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcaehbea-Jmf-CPhm3XM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Jmf-CPhm3XM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcaehbea-Jmf-CPhm3XM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843108&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gcaehbea-Jmf-CPhm3XM&#38;vembed=0&#38;video_id=Jmf-CPhm3XM&#38;video_target=tpm-plugin-gcaehbea-Jmf-CPhm3XM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Exponential+Expressions+Using+the+Quotient+Property+of+Exponents_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Exponential Expressions Using the Quotient Property of Exponents\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Raise Powers to Powers<\/h2>\n<p>Let\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is [latex]4[\/latex], so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).<\/p>\n<p>[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of [latex]5[\/latex] to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].<\/p>\n<p>So, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).<\/p>\n<p>Likewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]<\/p>\n<p>This leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].<\/p>\n<section class=\"textbox example\">Simplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q841688\">Show Solution<\/button><\/p>\n<div id=\"q841688\" class=\"hidden-answer\" style=\"display: none\">Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]6\\left(c^{4}\\right)^{2}=6c^{8}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Raise a Product to a Power<\/h2>\n<p>Simplify this expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\n<p>Notice that the exponent is applied to each factor of [latex]2a[\/latex]. So, we can eliminate the middle steps.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\n<p>The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdhfbfeg-Hgu9HKDHTUA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Hgu9HKDHTUA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fdhfbfeg-Hgu9HKDHTUA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843109&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fdhfbfeg-Hgu9HKDHTUA&#38;vembed=0&#38;video_id=Hgu9HKDHTUA&#38;video_target=tpm-plugin-fdhfbfeg-Hgu9HKDHTUA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Exponential+Expressions+Using+the+Power+Property+of+Exponents_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Exponential Expressions Using the Power Property of Exponents\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Raise a Quotient to a Power<\/h2>\n<p>Now let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex]\\displaystyle \\frac{3}{4}[\/latex] and raise it to the [latex]3[\/latex]<sup>rd<\/sup> power.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\n<p>You can see that raising the quotient to the power of [latex]3[\/latex] can also be written as the numerator ([latex]3[\/latex]) to the power of [latex]3[\/latex], and the denominator ([latex]4[\/latex]) to the power of [latex]3[\/latex].<\/p>\n<p>Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)=\\frac{a\\cdot a\\cdot a\\cdot a}{b\\cdot b\\cdot b\\cdot b}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\n<p>When a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bdgghedd-ZbxgDRV35dE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZbxgDRV35dE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bdgghedd-ZbxgDRV35dE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843110&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bdgghedd-ZbxgDRV35dE&#38;vembed=0&#38;video_id=ZbxgDRV35dE&#38;video_target=tpm-plugin-bdgghedd-ZbxgDRV35dE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Simplify+Fractions+Raised+to+Powers+(Positive+Exponents+Only)+Version+1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Zero Exponent Rule<\/h2>\n<h3>What if the exponent is zero?<\/h3>\n<p>To see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of [latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\n<p>If we were to simplify the original expression using the quotient rule, we would have<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/p>\n<p>If we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.<\/p>\n<p style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/p>\n<p>The sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or [latex]DNE[\/latex] (Does Not Exist).<\/p>\n<section class=\"textbox example\">Evaluate [latex]2x^{0}[\/latex] if [latex]x=9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q324798\">Show Solution<\/button><\/p>\n<div id=\"q324798\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]9[\/latex] for the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot9^{0}[\/latex]<\/p>\n<p>Evaluate [latex]9^{0}[\/latex]. Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot1=2[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]2x^{0}=2[\/latex], if [latex]x=9[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbhdfeeb-jKihp_DVDa0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/jKihp_DVDa0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbhdfeeb-jKihp_DVDa0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843126&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fbhdfeeb-jKihp_DVDa0&#38;vembed=0&#38;video_id=jKihp_DVDa0&#38;video_target=tpm-plugin-fbhdfeeb-jKihp_DVDa0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Evaluate+and+Simplify+Expressions+Using+the+Zero+Exponent+Rule_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate and Simplify Expressions Using the Zero Exponent Rule\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>The Negative Exponent Rule<\/h2>\n<p>Given a quotient like\u00a0[latex]\\displaystyle \\frac{{{2}^{m}}}{{{2}^{n}}}[\/latex] what happens when [latex]n[\/latex] is larger than [latex]m[\/latex]? We will need to use the <em>negative rule of exponents<\/em> to simplify the expression so that it is easier to understand.<\/p>\n<p>Let&#8217;s look at an example to clarify this idea. Given the expression:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{h}^{3}}{{h}^{5}}[\/latex]<\/p>\n<p>Expand the numerator and denominator, all the terms in the numerator will cancel to [latex]1[\/latex], leaving two [latex]h[\/latex]s multiplied in the denominator, and a numerator of [latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\frac{{h}^{3}}{{h}^{5}}\\,\\,\\,=\\,\\,\\,\\frac{h\\cdot{h}\\cdot{h}}{h\\cdot{h}\\cdot{h}\\cdot{h}\\cdot{h}} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}}{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}\\cdot {h}\\cdot {h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{h\\cdot{h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{{h}^{2}} \\end{array}[\/latex]<\/p>\n<p>We could have also applied the quotient rule from the last section, to obtain the following result:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{h^{3}}{h^{5}}\\,\\,\\,=\\,\\,\\,h^{3-5}\\\\\\\\=\\,\\,\\,h^{-2}\\,\\,\\end{array}[\/latex]<\/p>\n<p>Putting the answers together, we have [latex]{h}^{-2}=\\frac{1}{{h}^{2}}[\/latex]. This is true when [latex]h[\/latex], or any variable, is a real number and is not zero.<\/p>\n<section class=\"textbox example\">Write [latex]\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}[\/latex] with positive exponents.<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q219981\">Show Solution<\/button><\/p>\n<div id=\"q219981\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">Use the quotient rule to subtract the exponents of terms with like bases.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}={t}^{3-8}\\\\={t}^{-5}\\,\\,\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Write the expression with positive exponents by putting the term with the negative exponent in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{1}{{t}^{5}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]\\frac{1}{{t}^{5}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Simplify [latex]{\\left(\\frac{1}{3}\\right)}^{-2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q998337\">Show Solution<\/button><\/p>\n<div id=\"q998337\" class=\"hidden-answer\" style=\"display: none\">Apply the power property of exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{1}^{-2}}{{3}^{-2}}[\/latex]<\/p>\n<p>Write each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{3}^{2}}{{1}^{2}}{ = }\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}{ = }\\frac{9}{1}{ = }{9}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]9[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efaagccc-WvFlHjlIITg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/WvFlHjlIITg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-efaagccc-WvFlHjlIITg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843127&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-efaagccc-WvFlHjlIITg&#38;vembed=0&#38;video_id=WvFlHjlIITg&#38;video_target=tpm-plugin-efaagccc-WvFlHjlIITg\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Negative+Exponents+-+Basics_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Negative Exponents &#8211; Basics\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Convert Standard Notation to Scientific Notation<\/h2>\n<p>Remember working with place value for whole numbers and decimals? Our number system is based on powers of [latex]10[\/latex]. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens\u2014tenths, hundredths, thousandths, and so on.<\/p>\n<p>Consider the numbers [latex]4000[\/latex] and [latex]0.004[\/latex]. We know that [latex]4000[\/latex] means [latex]4\u00d71000[\/latex] and [latex]0.004[\/latex] means [latex]4 \u00d7 \\frac{1}{1000}[\/latex]. If we write the [latex]1000[\/latex] as a power of ten in exponential form, we can rewrite these numbers in this way:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc} \\hfill 4000 &&&&&& 0.004 \\hfill \\\\ 4 \\text{ x } 1000 &&&&&& 4 \\text{ x } \\frac{1}{1000} \\hfill \\\\ 4 \\text{ x } 10^3 &&&&&& 4 \\text{ x } \\frac{1}{10^3} \\hfill \\\\ &&&&&& 4 \\text{ x } 10^{-3} \\hfill \\\\ \\end{array}[\/latex]<\/p>\n<p>When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than [latex]10[\/latex], and the second factor is a power of [latex]10[\/latex] written in exponential form, it is said to be in scientific notation.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>a general note: scientific notation<\/h3>\n<p>A number is written in <strong>scientific notation<\/strong> if it is written in the form [latex]a\\times {10}^{n}[\/latex], where [latex]1\\le |a|<10[\/latex] and [latex]n[\/latex] is an integer.\n\n<\/div>\n<\/section>\n<p>Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier. If we look at what happened to the decimal point, we can see a method to easily convert from standard notation to scientific notation.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-166\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5838\/2022\/11\/28165215\/QR-Sci-Not-Dig-Deeper-1.jpg\" alt=\"On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, \u201cMoved the decimal point 3 places to the left.\u201d On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, \u201cMoved the decimal point 3 places to the right.\" width=\"449\" height=\"120\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>In both cases, the decimal was moved [latex]3[\/latex] places to get the first factor, [latex]4[\/latex], by itself.<\/p>\n<p>The power of [latex]10[\/latex] is positive when the number is larger than [latex]1[\/latex]: [latex]4000=4\u00d710^3[\/latex].<br \/>\nThe power of [latex]10[\/latex] is negative when the number is between [latex]0[\/latex] and [latex]1[\/latex]: [latex]0.004=4\u00d710^{-3}[\/latex].<\/p>\n<p>Watch the following video to see more examples of writing numbers in scientific notation.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-geacbdhc-fsNu3AdIgdk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/fsNu3AdIgdk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-geacbdhc-fsNu3AdIgdk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843128&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-geacbdhc-fsNu3AdIgdk&#38;vembed=0&#38;video_id=fsNu3AdIgdk&#38;video_target=tpm-plugin-geacbdhc-fsNu3AdIgdk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Examples+-+Write+a+Number+in+Scientific+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Write a Number in Scientific Notation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox seeExample\">Write each number in scientific notation.<\/p>\n<ol>\n<li>Distance to Andromeda Galaxy from Earth: [latex]24,000,000,000,000,000,000,000[\/latex] m<\/li>\n<li>Diameter of Andromeda Galaxy: [latex]1,300,000,000,000,000,000,000[\/latex] m<\/li>\n<li>Number of stars in Andromeda Galaxy: [latex]1,000,000,000,000[\/latex]<\/li>\n<li>Diameter of electron: [latex]0.00000000000094[\/latex] m<\/li>\n<li>Probability of being struck by lightning in any single year: [latex]0.00000143[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q500341\">Show Solution<\/button><\/p>\n<div id=\"q500341\" class=\"hidden-answer\" style=\"display: none\">\n<p>Observe that, if the given number is greater than [latex]1[\/latex], as in examples a\u2013c, the exponent of [latex]10[\/latex] is positive; and if the number is less than [latex]1[\/latex], as in examples d\u2013e, the exponent is negative.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2187\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2187&theme=lumen&iframe_resize_id=ohm2187&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Convert Scientific Notation to Standard Notation<\/h2>\n<p>How can we convert from scientific notation to standard notation? Let\u2019s look at two numbers written in scientific notation and see.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc} \\hfill 9.12 \\text{ x } 10^{4} &&&&&& 9.12 \\text{ x } 10^{-4} \\hfill \\\\ 9.12 \\text{ x } 10,000 &&&&&& 9.12 \\text{ x } 0.0001 \\hfill \\\\ 91,200 &&&&&& 0.000912 \\hfill \\\\ \\end{array}[\/latex]<\/div>\n<section class=\"textbox questionHelp\"><strong>How To: Convert Scientific Notation to Standard Notation<\/strong><\/p>\n<ul>\n<li>Step 1: Determine the exponent, [latex]n[\/latex], on the factor [latex]10[\/latex].<\/li>\n<li>Step 2: Move the decimal [latex]n[\/latex] places, adding zeros if needed.<\/li>\n<li>Step 3: Check your answer.<\/li>\n<\/ul>\n<\/section>\n<p>If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form \u2013 also known as standard notation.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-192\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5838\/2022\/11\/28175455\/5eea11b73e531ec9788f076d12a39212b3192193.jpg\" alt=\"On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.\" width=\"498\" height=\"65\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>In both cases, the decimal point moved [latex]4[\/latex] places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.<\/p>\n<p>Watch the following video to see more examples of writing scientific notation in standard notation.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aefgdhhd-8BX0oKUMIjw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/8BX0oKUMIjw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aefgdhhd-8BX0oKUMIjw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12539606&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aefgdhhd-8BX0oKUMIjw&#38;vembed=0&#38;video_id=8BX0oKUMIjw&#38;video_target=tpm-plugin-aefgdhhd-8BX0oKUMIjw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Examples+-+Writing+a+Number+in+Decimal+Notation+When+Given+in+Scientific+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Writing a Number in Decimal Notation When Given in Scientific Notation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox seeExample\">Convert each number in scientific notation to standard notation.<\/p>\n<ol>\n<li>[latex]3.547\\times {10}^{14}[\/latex]<\/li>\n<li>[latex]-2\\times {10}^{6}[\/latex]<\/li>\n<li>[latex]7.91\\times {10}^{-7}[\/latex]<\/li>\n<li>[latex]-8.05\\times {10}^{-12}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q500342\">Show Solution<\/button><\/p>\n<div id=\"q500342\" class=\"hidden-answer\" style=\"display: none\">\n1.<br \/>\n[latex]\\begin{align}&3.547\\times {10}^{14} \\\\ &\\underset{\\to 14\\text{ places}}{{3.54700000000000}} \\\\ &354,700,000,000,000 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>2.<br \/>\n[latex]\\begin{align}&-2\\times {10}^{6} \\\\ &\\underset{\\to 6\\text{ places}}{{-2.000000}} \\\\ &-2,000,000 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>3.<br \/>\n[latex]\\begin{align}&7.91\\times {10}^{-7} \\\\ &\\underset{\\to 7\\text{ places}}{{0000007.91}} \\\\ &0.000000791 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>4.<br \/>\n[latex]\\begin{align}&-8.05\\times {10}^{-12} \\\\ &\\underset{\\to 12\\text{ places}}{{-000000000008.05}} \\\\ &-0.00000000000805 \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2189\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2189&theme=lumen&iframe_resize_id=ohm2189&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We use the Properties of Exponents to multiply and divide numbers in scientific notation.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>multiplying and diving numbers in scientific notation<\/h3>\n<p>To multiply numbers in scientific notation, we need to multiply the coefficients and add the powers of [latex]10[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To divide numbers in scientific notation, we need to divide the coefficients and subtract the powers of [latex]10[\/latex].<\/p>\n<\/div>\n<\/section>\n<p>Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in [latex]1[\/latex] L of water. Each water molecule contains [latex]3[\/latex] atoms ([latex]2[\/latex] hydrogen and [latex]1[\/latex] oxygen). The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and [latex]1[\/latex] L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops. Therefore, there are approximately [latex]3\\cdot \\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in [latex]1[\/latex] L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!<\/p>\n<section class=\"textbox seeExample\">Perform the operations and write the answer in scientific notation.<\/p>\n<ol>\n<li>[latex]\\left(8.14\\times {10}^{-7}\\right)\\left(6.5\\times {10}^{10}\\right)[\/latex]<\/li>\n<li>[latex]\\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)[\/latex]<\/li>\n<li>[latex]\\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)[\/latex]<\/li>\n<li>[latex]\\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)[\/latex]<\/li>\n<li>[latex]\\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q500343\">Show Solution<\/button><\/p>\n<div id=\"q500343\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{align}\\left(8.14 \\times 10^{-7}\\right)\\left(6.5 \\times 10^{10}\\right) & =\\left(8.14 \\times 6.5\\right)\\left(10^{-7} \\times 10^{10}\\right) && \\text{Commutative and associative properties of multiplication} \\\\ & =\\left(52.91\\right)\\left(10^{3}\\right) && \\text{Product rule of exponents} \\\\ & =5.291 \\times 10^{4} && \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\left(4\\times {10}^{5}\\right)\\div \\left(-1.52\\times {10}^{9}\\right)& = \\left(\\frac{4}{-1.52}\\right)\\left(\\frac{{10}^{5}}{{10}^{9}}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(-2.63\\right)\\left({10}^{-4}\\right)&& \\text{Quotient rule of exponents} \\\\ & = -2.63\\times {10}^{-4}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\left(2.7\\times {10}^{5}\\right)\\left(6.04\\times {10}^{13}\\right)& = \\left(2.7\\times 6.04\\right)\\left({10}^{5}\\times {10}^{13}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(16.308\\right)\\left({10}^{18}\\right)&& \\text{Product rule of exponents} \\\\ & = 1.6308\\times {10}^{19}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\left(1.2\\times {10}^{8}\\right)\\div \\left(9.6\\times {10}^{5}\\right)& = \\left(\\frac{1.2}{9.6}\\right)\\left(\\frac{{10}^{8}}{{10}^{5}}\\right)&& \\text{Commutative and associative properties of multiplication} \\\\ & = \\left(0.125\\right)\\left({10}^{3}\\right)&& \\text{Quotient rule of exponents} \\\\ & = 1.25\\times {10}^{2}&& \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\left(3.33\\times {10}^{4}\\right)\\left(-1.05\\times {10}^{7}\\right)\\left(5.62\\times {10}^{5}\\right)& = \\left[3.33\\times \\left(-1.05\\right)\\times 5.62\\right]\\left({10}^{4}\\times {10}^{7}\\times {10}^{5}\\right) \\\\ & \\approx \\left(-19.65\\right)\\left({10}^{16}\\right) \\\\ & = -1.965\\times {10}^{17} \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video to see more examples of multiplying and dividing numbers in scientific notation.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-beegacfg-yX6Mq9whsX0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/yX6Mq9whsX0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-beegacfg-yX6Mq9whsX0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=10294935&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-beegacfg-yX6Mq9whsX0&#38;vembed=0&#38;video_id=yX6Mq9whsX0&#38;video_target=tpm-plugin-beegacfg-yX6Mq9whsX0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Scientific+Notation+-+Multiplication+and+Division_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cScientific Notation &#8211; Multiplication and Division\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2190\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2190&theme=lumen&iframe_resize_id=ohm2190&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox seeExample\">An average human body contains around [latex]30,000,000,000,000[\/latex] red blood cells. Each cell measures approximately [latex]0.000008[\/latex] m long. Find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q500344\">Show Solution<\/button><\/p>\n<div id=\"q500344\" class=\"hidden-answer\" style=\"display: none\">First, we must write both the number of red blood cells and the length in scientific notation.Number of red blood cells:[latex]30,000,000,000,000[\/latex] = [latex]3\\times {10}^{13}[\/latex]<em>Note: In order to get [latex]3[\/latex] by itself we moved the decimal place [latex]13[\/latex] times to the left making our exponent positive [latex]13[\/latex].<\/em>Length of red blood cells:[latex]0.000008 = 8\\times {10}^{-6}[\/latex]<br \/>\n<em>Note: In order to get [latex]8[\/latex] by itself we moved the decimal place [latex]6[\/latex] times to the right making our exponent negative [latex]6[\/latex].<\/em> Now that we have both numbers in scientific notation, we can find the total length of all the cells laid end-to-end.To find the total length we must multiple the number of red blood cells by the length of each blood cell.[latex]\\begin{align}\\left(3 \\times 10^{13}\\right)\\left(8 \\times 10^{-6}\\right) & =\\left(3 \\times 8\\right)\\left(10^{13} \\times 10^{-6}\\right) && \\text{Commutative and associative properties of multiplication} \\\\ & =\\left(24\\right)\\left(10^{7}\\right) && \\text{Product rule of exponents} \\\\ & =2.4 \\times 10^{8} && \\text{Scientific notation} \\\\ \\text{ } \\end{align}[\/latex]Number of cells: [latex]2.4\\times {10}^{8}[\/latex] m or [latex]240,000,000[\/latex] m.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1842\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1842&theme=lumen&iframe_resize_id=ohm1842&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=Jmf-CPhm3XM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1 \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=ZbxgDRV35dE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Evaluate and Simplify Expressions Using the Zero Exponent Rule 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