{"id":611,"date":"2024-04-23T18:31:24","date_gmt":"2024-04-23T18:31:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=611"},"modified":"2024-11-20T00:53:01","modified_gmt":"2024-11-20T00:53:01","slug":"exponents-and-scientific-notation-learn-it-7","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponents-and-scientific-notation-learn-it-7\/","title":{"raw":"Exponents and Scientific Notation: Learn It 7","rendered":"Exponents and Scientific Notation: Learn It 7"},"content":{"raw":"<h2>Using Scientific Notation in Applications<\/h2>\r\nNow that we've mastered converting between standard and scientific notation, let\u2019s explore how this skill can be applied in real-world scenarios.\r\n\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18744[\/ohm2_question]<\/section><section>Let's dive into some practical exercises to see scientific notation in action, helping us solve problems efficiently in science, engineering, and beyond.<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18739[\/ohm2_question]<\/section><section class=\"textbox example\">Suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen).The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and 1 L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops.Therefore, there are approximately [latex]\\left(3\\right)\\cdot\\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)=\\left(3\\cdot1.32\\cdot1.22\\right)\\times\\left({10}^{4}\\cdot{10}^{25}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in 1 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\nNote: How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this?\r\n<p style=\"padding-left: 40px;\">Recall that multiplication is both [pb_glossary id=\"615\"]commutative[\/pb_glossary] and [pb_glossary id=\"616\"]associative[\/pb_glossary]. That means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the [pb_glossary id=\"617\"]product rule for exponents[\/pb_glossary] allows us to add the exponents on the base of [latex]10[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\">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=\"380183\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"380183\"]\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><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18747[\/ohm2_question]<\/section>","rendered":"<h2>Using Scientific Notation in Applications<\/h2>\n<p>Now that we&#8217;ve mastered converting between standard and scientific notation, let\u2019s explore how this skill can be applied in real-world scenarios.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18744\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18744&theme=lumen&iframe_resize_id=ohm18744&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>Let&#8217;s dive into some practical exercises to see scientific notation in action, helping us solve problems efficiently in science, engineering, and beyond.<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18739\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18739&theme=lumen&iframe_resize_id=ohm18739&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen).The average drop of water contains around [latex]1.32\\times {10}^{21}[\/latex] molecules of water and 1 L of water holds about [latex]1.22\\times {10}^{4}[\/latex] average drops.Therefore, there are approximately [latex]\\left(3\\right)\\cdot\\left(1.32\\times {10}^{21}\\right)\\cdot \\left(1.22\\times {10}^{4}\\right)=\\left(3\\cdot1.32\\cdot1.22\\right)\\times\\left({10}^{4}\\cdot{10}^{25}\\right)\\approx 4.83\\times {10}^{25}[\/latex] atoms in 1 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<p>Note: How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this?<\/p>\n<p style=\"padding-left: 40px;\">Recall that multiplication is both <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_611_615\">commutative<\/a> and <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_611_616\">associative<\/a>. That means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_611_617\">product rule for exponents<\/a> allows us to add the exponents on the base of [latex]10[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">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=\"q380183\">Show Answer<\/button><\/p>\n<div id=\"q380183\" 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<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18747\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18747&theme=lumen&iframe_resize_id=ohm18747&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_611_615\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_611_615\"><div tabindex=\"-1\"><p>Commutative Property of Multiplication states that the order in which numbers are multiplied does not affect the product.<br \/>\nThat is: [latex]a \\cdot b = b \\cdot a[\/latex]<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_611_616\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_611_616\"><div tabindex=\"-1\"><p>Associative Property of Multiplication indicates that when three or more numbers are multiplied, the grouping of the numbers does not change the product.<br \/>\nIn mathematical terms, [latex](a \\cdot b) \\cdot c = a \\cdot (b \\cdot c)[\/latex].<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_611_617\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_611_617\"><div tabindex=\"-1\"><p>The product rule for exponents states that when you multiply two powers of the same base, you can add their exponents. So, [latex]a^m \\cdot a^n = a^{m+n}[\/latex].<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":32,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/611"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/611\/revisions"}],"predecessor-version":[{"id":6217,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/611\/revisions\/6217"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/32"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/611\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=611"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=611"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=611"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=611"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}