{"id":15341,"date":"2021-10-11T23:05:34","date_gmt":"2021-10-11T23:05:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/summary-simplifying-expressions-with-negative-exponents\/"},"modified":"2021-10-17T19:55:36","modified_gmt":"2021-10-17T19:55:36","slug":"summary-exponents-scientific-notation-and-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/summary-exponents-scientific-notation-and-square-roots\/","title":{"raw":"Summary: Exponents, Scientific Notation and Square Roots","rendered":"Summary: Exponents, Scientific Notation and Square Roots"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li><strong>Exponential Notation<\/strong>\r\n<ul id=\"eip-277\"><\/ul>\r\n<\/li>\r\n \t<li style=\"list-style-type: none;\">\r\n<p style=\"padding-left: 150px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224415\/CNX_BMath_Figure_10_02_026_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" \/>\r\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.<\/p>\r\n<\/li>\r\n \t<li><strong>Properties of Exponent<\/strong>\r\n<ul id=\"eip-id1170324231481\">\r\n \t<li>If [latex]a,b[\/latex] are real numbers and [latex]m,n[\/latex] are integers, then\r\n[latex]\\begin{array}{cccc}\\mathbf{\\text{Product Property}}\\hfill &amp; &amp; &amp; {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\mathbf{\\text{Power Property}}\\hfill &amp; &amp; &amp; {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\mathbf{\\text{Power of a Product Property}}\\hfill &amp; &amp; &amp; {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\\\ \\mathbf{\\text{Quotient Property}}\\hfill &amp; &amp; &amp; {\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},a\\ne 0\\hfill \\\\ \\mathbf{\\text{Zero Exponent Property}}\\hfill &amp; &amp; &amp; {a}^{0}=1,a\\ne 0\\hfill \\\\ \\mathbf{\\text{Power of a Quotient Property}}\\hfill &amp; &amp; &amp; {\\left({\\Large\\frac{a}{b}}\\right)}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}},b\\ne 0\\hfill \\\\ \\mathbf{\\text{Definition of Negative Exponent}}\\hfill &amp; &amp; &amp; {a}^{-n}={\\Large\\frac{1}{{a}^{n}}}\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Convert from Decimal Notation to Scientific Notation:<\/strong> To convert a decimal to scientific notation:\r\n<ol id=\"eip-id1170326348637\" class=\"stepwise\">\r\n \t<li>Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.<\/li>\r\n \t<li>Count the number of decimal places, [latex]n[\/latex] , that the decimal point was moved.<\/li>\r\n \t<li>Write the number as a product with a power of [latex]10[\/latex].\r\n<ul id=\"eip-id1170326348647\">\r\n \t<li>If the original number is greater than [latex]1[\/latex], the power of [latex]10[\/latex] will be [latex]{10}^{n}[\/latex] .<\/li>\r\n \t<li>If the original number is between [latex]0[\/latex] and [latex]1[\/latex], the power of [latex]10[\/latex] will be [latex]{10}^{n}[\/latex] .<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Check.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Convert from Scientific Notation to Decimal Notation:<\/strong> To convert scientific notation to decimal form:\r\n<ol id=\"eip-id1170323926258\" class=\"stepwise\">\r\n \t<li>Determine the exponent, [latex]n[\/latex] , on the factor [latex]10[\/latex].<\/li>\r\n \t<li>Move the decimal [latex]n[\/latex] places, adding zeros if needed.\r\n<ul id=\"eip-id1170323926264\">\r\n \t<li>If the exponent is positive, move the decimal point [latex]n[\/latex] places to the right.<\/li>\r\n \t<li>If the exponent is negative, move the decimal point [latex]|n|[\/latex] places to the left.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Check.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Square Root Notation<\/strong> [latex]\\sqrt{m}[\/latex] is read \u2018the square root of [latex]m[\/latex] \u2019.\u00a0 If [latex]m={n}^{2}[\/latex] , then [latex]\\sqrt{m}=n[\/latex] , for [latex]n\\ge 0[\/latex] .\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221847\/CNX_BMath_Figure_05_07_011_img.png\" alt=\".\" \/><\/li>\r\n \t<li><strong>Square Roots and Area<\/strong> If the area of the square is A square units, the length of a side is [latex]\\sqrt{A}[\/latex] units.<\/li>\r\n \t<li><strong>Square Roots and Gravity<\/strong>\u00a0On Earth, if an object is dropped from a height of [latex]h[\/latex]\u00a0feet,\u00a0the time in seconds it will take to reach the ground is found by evaluating the expression\u00a0[latex]{\\Large\\frac{\\sqrt{h}}{4}}[\/latex].<\/li>\r\n \t<li><strong>Square Roots and Accident Investigations<\/strong> Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is [latex]d[\/latex] feet, then the speed of the car can be found by evaluating\u00a0[latex]\\sqrt{24d}[\/latex].\r\n.<\/li>\r\n \t<li><strong>Use a strategy for applications with square roots.<\/strong>\r\n<ul id=\"eip-id1170325410660\">\r\n \t<li>Identify what you are asked to find.<\/li>\r\n \t<li>Write a phrase that gives the information to find it.<\/li>\r\n \t<li>Translate the phrase to an expression.<\/li>\r\n \t<li>Simplify the expression.<\/li>\r\n \t<li>Write a complete sentence that answers the question.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1529452\" class=\"definition\">\r\n \t<dt class=\"no-emphasis\"><strong>negative exponent<\/strong><\/dt>\r\n \t<dd id=\"fs-id1529458\">If [latex]n[\/latex] is a positive integer and [latex]a\\ne 0[\/latex] , then [latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex] .<\/dd>\r\n \t<dd><\/dd>\r\n \t<dt><strong>scientific notation<\/strong><\/dt>\r\n \t<dd>A number expressed in the form [latex]a\\times {10}^{n}[\/latex], where [latex]a\\ge 1[\/latex] and [latex]a&lt;10[\/latex], and [latex]n[\/latex] is an integer.<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>perfect square<\/dt>\r\n \t<dd id=\"fs-id2447033\">A perfect square is the square of a whole number.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id2946118\" class=\"definition\">\r\n \t<dt>square root of a number<\/dt>\r\n \t<dd id=\"fs-id2447033\">A number whose square is [latex]m[\/latex] is called a square root of [latex]m[\/latex].\r\nIf [latex]{n}^{2}=m[\/latex], then [latex]n[\/latex] is a square root of [latex]m[\/latex].<\/dd>\r\n<\/dl>\r\n<h2><\/h2>\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1XgOd4QvDdM5SgW0SXq1OZPYGZACrQN1w1Mk_7bIMnGk\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li><strong>Exponential Notation<\/strong>\n<ul id=\"eip-277\"><\/ul>\n<\/li>\n<li style=\"list-style-type: none;\">\n<p style=\"padding-left: 150px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224415\/CNX_BMath_Figure_10_02_026_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" \/><br \/>\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.<\/p>\n<\/li>\n<li><strong>Properties of Exponent<\/strong>\n<ul id=\"eip-id1170324231481\">\n<li>If [latex]a,b[\/latex] are real numbers and [latex]m,n[\/latex] are integers, then<br \/>\n[latex]\\begin{array}{cccc}\\mathbf{\\text{Product Property}}\\hfill & & & {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\mathbf{\\text{Power Property}}\\hfill & & & {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\mathbf{\\text{Power of a Product Property}}\\hfill & & & {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\\\ \\mathbf{\\text{Quotient Property}}\\hfill & & & {\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},a\\ne 0\\hfill \\\\ \\mathbf{\\text{Zero Exponent Property}}\\hfill & & & {a}^{0}=1,a\\ne 0\\hfill \\\\ \\mathbf{\\text{Power of a Quotient Property}}\\hfill & & & {\\left({\\Large\\frac{a}{b}}\\right)}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}},b\\ne 0\\hfill \\\\ \\mathbf{\\text{Definition of Negative Exponent}}\\hfill & & & {a}^{-n}={\\Large\\frac{1}{{a}^{n}}}\\hfill \\end{array}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Convert from Decimal Notation to Scientific Notation:<\/strong> To convert a decimal to scientific notation:\n<ol id=\"eip-id1170326348637\" class=\"stepwise\">\n<li>Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.<\/li>\n<li>Count the number of decimal places, [latex]n[\/latex] , that the decimal point was moved.<\/li>\n<li>Write the number as a product with a power of [latex]10[\/latex].\n<ul id=\"eip-id1170326348647\">\n<li>If the original number is greater than [latex]1[\/latex], the power of [latex]10[\/latex] will be [latex]{10}^{n}[\/latex] .<\/li>\n<li>If the original number is between [latex]0[\/latex] and [latex]1[\/latex], the power of [latex]10[\/latex] will be [latex]{10}^{n}[\/latex] .<\/li>\n<\/ul>\n<\/li>\n<li>Check.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Convert from Scientific Notation to Decimal Notation:<\/strong> To convert scientific notation to decimal form:\n<ol id=\"eip-id1170323926258\" class=\"stepwise\">\n<li>Determine the exponent, [latex]n[\/latex] , on the factor [latex]10[\/latex].<\/li>\n<li>Move the decimal [latex]n[\/latex] places, adding zeros if needed.\n<ul id=\"eip-id1170323926264\">\n<li>If the exponent is positive, move the decimal point [latex]n[\/latex] places to the right.<\/li>\n<li>If the exponent is negative, move the decimal point [latex]|n|[\/latex] places to the left.<\/li>\n<\/ul>\n<\/li>\n<li>Check.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Square Root Notation<\/strong> [latex]\\sqrt{m}[\/latex] is read \u2018the square root of [latex]m[\/latex] \u2019.\u00a0 If [latex]m={n}^{2}[\/latex] , then [latex]\\sqrt{m}=n[\/latex] , for [latex]n\\ge 0[\/latex] .<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221847\/CNX_BMath_Figure_05_07_011_img.png\" alt=\".\" \/><\/li>\n<li><strong>Square Roots and Area<\/strong> If the area of the square is A square units, the length of a side is [latex]\\sqrt{A}[\/latex] units.<\/li>\n<li><strong>Square Roots and Gravity<\/strong>\u00a0On Earth, if an object is dropped from a height of [latex]h[\/latex]\u00a0feet,\u00a0the time in seconds it will take to reach the ground is found by evaluating the expression\u00a0[latex]{\\Large\\frac{\\sqrt{h}}{4}}[\/latex].<\/li>\n<li><strong>Square Roots and Accident Investigations<\/strong> Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is [latex]d[\/latex] feet, then the speed of the car can be found by evaluating\u00a0[latex]\\sqrt{24d}[\/latex].<br \/>\n.<\/li>\n<li><strong>Use a strategy for applications with square roots.<\/strong>\n<ul id=\"eip-id1170325410660\">\n<li>Identify what you are asked to find.<\/li>\n<li>Write a phrase that gives the information to find it.<\/li>\n<li>Translate the phrase to an expression.<\/li>\n<li>Simplify the expression.<\/li>\n<li>Write a complete sentence that answers the question.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1529452\" class=\"definition\">\n<dt class=\"no-emphasis\"><strong>negative exponent<\/strong><\/dt>\n<dd id=\"fs-id1529458\">If [latex]n[\/latex] is a positive integer and [latex]a\\ne 0[\/latex] , then [latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex] .<\/dd>\n<dd><\/dd>\n<dt><strong>scientific notation<\/strong><\/dt>\n<dd>A number expressed in the form [latex]a\\times {10}^{n}[\/latex], where [latex]a\\ge 1[\/latex] and [latex]a<10[\/latex], and [latex]n[\/latex] is an integer.<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>perfect square<\/dt>\n<dd id=\"fs-id2447033\">A perfect square is the square of a whole number.<\/dd>\n<\/dl>\n<dl id=\"fs-id2946118\" class=\"definition\">\n<dt>square root of a number<\/dt>\n<dd id=\"fs-id2447033\">A number whose square is [latex]m[\/latex] is called a square root of [latex]m[\/latex].<br \/>\nIf [latex]{n}^{2}=m[\/latex], then [latex]n[\/latex] is a square root of [latex]m[\/latex].<\/dd>\n<\/dl>\n<h2><\/h2>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1XgOd4QvDdM5SgW0SXq1OZPYGZACrQN1w1Mk_7bIMnGk\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15341\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":167848,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"070e1afdb71d4fe699174d113cbbd5c3","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15341","chapter","type-chapter","status-publish","hentry"],"part":15430,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15341\/revisions"}],"predecessor-version":[{"id":15679,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15341\/revisions\/15679"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/15430"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15341\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15341"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15341"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15341"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}