{"id":1444,"date":"2023-06-22T02:28:38","date_gmt":"2023-06-22T02:28:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-16-background-youll-need-5\/"},"modified":"2024-03-01T20:09:37","modified_gmt":"2024-03-01T20:09:37","slug":"module-16-background-youll-need-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-16-background-youll-need-5\/","title":{"raw":"Module 15: Background You'll Need 5","rendered":"Module 15: Background You&#8217;ll Need 5"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Simplify exponents&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12929,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:9}\">Simplify exponents<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>exponents<\/h3>\r\n<p>The mathematical operation of exponentiation is denoted using superscript notation:<\/p>\r\n<p style=\"text-align: center;\">[latex]b^x[\/latex]<\/p>\r\n<p>Some ways that this can be read include \u201c[latex]b[\/latex]\u00a0raised to the power of [latex]x[\/latex]\u201d or \u201c[latex]b[\/latex]\u00a0raised to the [latex]x[\/latex]\u00a0power.\u201d<\/p>\r\n<p>The quantity [latex]b[\/latex]\u00a0is called the <strong>base<\/strong>, and the quantity [latex]x[\/latex]\u00a0is called the <strong>exponent<\/strong>.<\/p>\r\n<p>When the exponent is a positive integer, the exponent describes how many times to multiply the base by itself.<\/p>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox example\">When the exponent is 2, we say that we are <strong>squaring<\/strong> the base. The quantity\r\n\r\n<p style=\"text-align: center;\">[latex]b^2[\/latex]<\/p>\r\n<p>can be read as \u201c[latex]b[\/latex]\u00a0raised to the power of [latex]2[\/latex]\u201d or \u201c[latex]b[\/latex]\u00a0raised to the second power,\u201d as described above, but it can also be read as \u201c[latex]b[\/latex]\u00a0squared.\u201d<\/p>\r\n<p>For example: [latex]4^2 = 4 \\cdot 4 = 16[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">When the exponent is 3, we say that we are <strong>cubing<\/strong> the base. The quantity\r\n\r\n<p style=\"text-align: center;\">[latex]b^3[\/latex]<\/p>\r\n<p>can be read as \u201c[latex]b[\/latex]\u00a0raised to the power of [latex]3[\/latex]\u201d or \u201c[latex]b[\/latex]\u00a0raised to the third power,\u201d as described above, but it can also be read as \u201c[latex]b[\/latex]\u00a0cubed.\u201d<\/p>\r\n<p>For example: [latex]4^3 = 4 \\cdot 4 \\cdot 4 = 64[\/latex]<\/p>\r\n<\/section>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox example\">Calculate [latex]2^4[\/latex].<br \/>\r\nNotice that the base is [latex]2[\/latex] and the exponent is [latex]4[\/latex].<br \/>\r\n[latex]4[\/latex] describes how many times to multiply the base [latex]2[\/latex] by itself.[latex]2^4 = 2 \\cdot 2 \\cdot 2 \\cdot 2 = 16[\/latex]<\/section>\r\n<\/section>\r\n<section class=\"textbox problemSolving\">What you have discovered is that when an exponent is negative, it tells us to take the reciprocal of the result we get when we have a positive exponent. In other words,\r\n\r\n<p style=\"text-align: center;\">[latex]b^{-x} = \\dfrac{1}{b^x}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3197[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Simplify exponents&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12929,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:9}\">Simplify exponents<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>exponents<\/h3>\n<p>The mathematical operation of exponentiation is denoted using superscript notation:<\/p>\n<p style=\"text-align: center;\">[latex]b^x[\/latex]<\/p>\n<p>Some ways that this can be read include \u201c[latex]b[\/latex]\u00a0raised to the power of [latex]x[\/latex]\u201d or \u201c[latex]b[\/latex]\u00a0raised to the [latex]x[\/latex]\u00a0power.\u201d<\/p>\n<p>The quantity [latex]b[\/latex]\u00a0is called the <strong>base<\/strong>, and the quantity [latex]x[\/latex]\u00a0is called the <strong>exponent<\/strong>.<\/p>\n<p>When the exponent is a positive integer, the exponent describes how many times to multiply the base by itself.<\/p>\n<\/section>\n<section>\n<section class=\"textbox example\">When the exponent is 2, we say that we are <strong>squaring<\/strong> the base. The quantity<\/p>\n<p style=\"text-align: center;\">[latex]b^2[\/latex]<\/p>\n<p>can be read as \u201c[latex]b[\/latex]\u00a0raised to the power of [latex]2[\/latex]\u201d or \u201c[latex]b[\/latex]\u00a0raised to the second power,\u201d as described above, but it can also be read as \u201c[latex]b[\/latex]\u00a0squared.\u201d<\/p>\n<p>For example: [latex]4^2 = 4 \\cdot 4 = 16[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">When the exponent is 3, we say that we are <strong>cubing<\/strong> the base. The quantity<\/p>\n<p style=\"text-align: center;\">[latex]b^3[\/latex]<\/p>\n<p>can be read as \u201c[latex]b[\/latex]\u00a0raised to the power of [latex]3[\/latex]\u201d or \u201c[latex]b[\/latex]\u00a0raised to the third power,\u201d as described above, but it can also be read as \u201c[latex]b[\/latex]\u00a0cubed.\u201d<\/p>\n<p>For example: [latex]4^3 = 4 \\cdot 4 \\cdot 4 = 64[\/latex]<\/p>\n<\/section>\n<\/section>\n<section>\n<section class=\"textbox example\">Calculate [latex]2^4[\/latex].<br \/>\nNotice that the base is [latex]2[\/latex] and the exponent is [latex]4[\/latex].<br \/>\n[latex]4[\/latex] describes how many times to multiply the base [latex]2[\/latex] by itself.[latex]2^4 = 2 \\cdot 2 \\cdot 2 \\cdot 2 = 16[\/latex]<\/section>\n<\/section>\n<section class=\"textbox problemSolving\">What you have discovered is that when an exponent is negative, it tells us to take the reciprocal of the result we get when we have a positive exponent. In other words,<\/p>\n<p style=\"text-align: center;\">[latex]b^{-x} = \\dfrac{1}{b^x}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3197\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3197&theme=lumen&iframe_resize_id=ohm3197&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1438,"module-header":"background_you_need","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\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1444"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1444\/revisions"}],"predecessor-version":[{"id":5854,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1444\/revisions\/5854"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1438"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1444\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1444"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1444"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1444"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}