{"id":16232,"date":"2019-10-01T22:23:46","date_gmt":"2019-10-01T22:23:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-multiply-and-divide-numbers-in-scientific-notation\/"},"modified":"2024-04-30T21:32:57","modified_gmt":"2024-04-30T21:32:57","slug":"read-multiply-and-divide-numbers-in-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-multiply-and-divide-numbers-in-scientific-notation\/","title":{"raw":"Multiplying and Dividing Numbers in Scientific Notation","rendered":"Multiplying and Dividing Numbers in Scientific Notation"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply numbers expressed in\u00a0scientific notation<\/li>\r\n \t<li>Divide numbers expressed in scientific notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title2\">Multiplying Numbers Expressed in Scientific Notation<\/h2>\r\nNumbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of the properties of numbers and the rules of exponents that you may recall. To multiply numbers in scientific notation, first multiply the numbers that aren\u2019t powers of 10 (the <i>a<\/i> in [latex]a\\times10^{n}[\/latex]). Then multiply the powers of ten by adding the exponents.\r\n\r\nThis will produce a new number times a different power of [latex]10[\/latex]. All you have to do is check to make sure this new value is in scientific notation. If it isn\u2019t, you convert it.\r\n\r\nLet\u2019s look at some examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\left(3\\times10^{8}\\right)\\left(6.8\\times10^{-13}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"395606\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"395606\"]Regroup using the commutative and associative properties.\r\n<p style=\"text-align: center;\">[latex]\\left(3\\times6.8\\right)\\left(10^{8}\\times10^{-13}\\right)[\/latex]<\/p>\r\nMultiply the coefficients.\r\n<p style=\"text-align: center;\">[latex]\\left(20.4\\right)\\left(10^{8}\\times10^{-13}\\right)[\/latex]<\/p>\r\nMultiply the powers of [latex]10[\/latex] using the Product Rule. Add the exponents.\r\n<p style=\"text-align: center;\">[latex]20.4\\times10^{-5}[\/latex]<\/p>\r\nConvert [latex]20.4[\/latex] into scientific notation by moving the decimal point one place to the left and multiplying by [latex]10^{1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(2.04\\times10^{1}\\right)\\times10^{-5}[\/latex]<\/p>\r\nGroup the powers of [latex]10[\/latex] using the associative property of multiplication.\r\n<p style=\"text-align: center;\">[latex]2.04\\times\\left(10^{1}\\times10^{-5}\\right)[\/latex]<\/p>\r\nMultiply using the Product Rule\u2014add the exponents.\r\n<p style=\"text-align: center;\">[latex]2.04\\times10^{1+\\left(-5\\right)}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3\\times10^{8}\\right)\\left(6.8\\times10^{-13}\\right)=2.04\\times10^{-4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\left(8.2\\times10^{6}\\right)\\left(1.5\\times10^{-3}\\right)\\left(1.9\\times10^{-7}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"23947\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23947\"]Regroup using the commutative and associative properties.\r\n<p style=\"text-align: center;\">[latex]\\left(8.2\\times1.5\\times1.9\\right)\\left(10^{6}\\times10^{-3}\\times10^{-7}\\right)[\/latex]<\/p>\r\nMultiply the numbers.\r\n<p style=\"text-align: center;\">[latex]\\left(23.37\\right)\\left(10^{6}\\times10^{-3}\\times10^{-7}\\right)[\/latex]<\/p>\r\nMultiply the powers of [latex]10[\/latex] using the Product Rule\u2014add the exponents.\r\n<p style=\"text-align: center;\">[latex]23.37\\times10^{-4}[\/latex]<\/p>\r\nConvert [latex]23.37[\/latex] into scientific notation by moving the decimal point one place to the left and multiplying by [latex]10^{1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(2.337\\times10^{1}\\right)\\times10^{-4}[\/latex]<\/p>\r\nGroup the powers of [latex]10[\/latex] using the associative property of multiplication.\r\n<p style=\"text-align: center;\">[latex]2.337\\times\\left(10^{1}\\times10^{-4}\\right)[\/latex]<\/p>\r\nMultiply using the Product Rule and add the exponents.\r\n<p style=\"text-align: center;\">[latex]2.337\\times10^{1+\\left(-4\\right)}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(8.2\\times10^{6}\\right)\\left(1.5\\times10^{-3}\\right)\\left(1.9\\times10^{-7}\\right)=2.337\\times10^{-3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply. Write answers in decimal form: [latex]\\left(4\\times {10}^{5}\\right)\\left(2\\times {10}^{-7}\\right)[\/latex].\r\n[reveal-answer q=\"630924\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"630924\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469641319\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(4\\times {10}^{5}\\right)\\left(2\\times {10}^{-7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to rearrange the factors.<\/td>\r\n<td>[latex]4\\cdot 2\\cdot {10}^{5}\\cdot {10}^{-7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]4[\/latex] by [latex]2[\/latex] and use the Product Property to multiply [latex]{10}^{5}[\/latex] by [latex]{10}^{-7}[\/latex].<\/td>\r\n<td>[latex]8\\times {10}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change to decimal form by moving the decimal two places left.<\/td>\r\n<td>[latex]{\\Large\\frac{8}{100}} = 0.08[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146318[\/ohm_question]\r\n\r\n[ohm_question]2826[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the following video you will see an example of how to multiply tow numbers that are written in scientific notation.\r\n\r\nhttps:\/\/youtu.be\/5ZAY4OCkp7U\r\n<h2 id=\"title2\">Dividing Numbers Expressed in Scientific Notation<\/h2>\r\nIn order to divide numbers in scientific notation, you once again apply the properties of numbers and the rules of exponents. You begin by dividing the numbers that aren\u2019t powers of [latex]10[\/latex] (the <i>a<\/i> in [latex]a\\times10^{n}[\/latex]. Then you divide the powers of ten by subtracting the exponents.\r\n\r\nThis will produce a new number times a different power of 10. If it isn\u2019t already in scientific notation, you convert it, and then you\u2019re done.\r\n\r\nLet\u2019s look at some examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{2.829\\times 1{{0}^{-9}}}{3.45\\times 1{{0}^{-3}}}[\/latex]<\/p>\r\n[reveal-answer q=\"364796\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"364796\"]Regroup using the associative property.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left( \\frac{2.829}{3.45} \\right)\\left( \\frac{{{10}^{-9}}}{{{10}^{-3}}} \\right)[\/latex]<\/p>\r\nDivide the coefficients.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left(0.82\\right)\\left( \\frac{{{10}^{-9}}}{{{10}^{-3}}} \\right)[\/latex]<\/p>\r\nDivide the powers of [latex]10[\/latex] using the Quotient Rule. Subtract the exponents.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0.82\\times10^{-9-\\left(-3\\right)}\\\\0.82\\times10^{-6}\\end{array}[\/latex]<\/p>\r\nConvert [latex]0.82[\/latex] into scientific notation by moving the decimal point one place to the right and multiplying by [latex]10^{-1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(8.2\\times10^{-1}\\right)\\times10^{-6}[\/latex]<\/p>\r\nGroup the powers of [latex]10[\/latex] together using the associative property.\r\n<p style=\"text-align: center;\">[latex]8.2\\times\\left(10^{-1}\\times10^{-6}\\right)[\/latex]<\/p>\r\nMultiply the powers of [latex]10[\/latex] using the Product Rule\u2014add the exponents.\r\n<p style=\"text-align: center;\">[latex]8.2\\times10^{-1+\\left(-6\\right)}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{2.829\\times {{10}^{-9}}}{3.45\\times {{10}^{-3}}}=8.2\\times {{10}^{-7}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{\\left(1.37\\times10^{4}\\right)\\left(9.85\\times10^{6}\\right)}{5.0\\times10^{12}}[\/latex]<\/p>\r\n[reveal-answer q=\"337143\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"337143\"]Regroup the terms in the numerator according to the associative and commutative properties.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{\\left( 1.37\\times 9.85 \\right)\\left( {{10}^{6}}\\times {{10}^{4}} \\right)}{5.0\\times {{10}^{12}}}[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{13.4945\\times {{10}^{10}}}{5.0\\times {{10}^{12}}}[\/latex]<\/p>\r\nRegroup using the associative property.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left( \\frac{13.4945}{5.0} \\right)\\left( \\frac{{{10}^{10}}}{{{10}^{12}}} \\right)[\/latex]<\/p>\r\nDivide the numbers.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left(2.6989\\right)\\left(\\frac{10^{10}}{10^{12}}\\right)[\/latex]<\/p>\r\nDivide the powers of [latex]10[\/latex] using the Quotient Rule\u2014subtract the exponents.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{c}\\left(2.6989 \\right)\\left( {{10}^{10-12}} \\right)\\\\2.6989\\times {{10}^{-2}}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{\\left( 1.37\\times {{10}^{4}} \\right)\\left( 9.85\\times {{10}^{6}} \\right)}{5.0\\times {{10}^{12}}}=2.6989\\times {{10}^{-2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide. Write answers in decimal form: [latex]{\\Large\\frac{9\\times {10}^{3}}{3\\times {10}^{-2}}}[\/latex].\r\n[reveal-answer q=\"424217\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"424217\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469868577\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{9\\times {10}^{3}}{3\\times {10}^{-2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Separate the factors.<\/td>\r\n<td>[latex]{\\Large\\frac{9}{3}}\\times {\\Large\\frac{{10}^{3}}{{10}^{-2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]9[\/latex] by [latex]3[\/latex] and use the Quotient Property to divide [latex]{10}^{3}[\/latex] by [latex]{10}^{-2}[\/latex] .<\/td>\r\n<td>[latex]3\\times {10}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change to decimal form by moving the decimal five places right.<\/td>\r\n<td>[latex]300,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146319[\/ohm_question]\r\n[ohm_question]2829[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that when you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator. You will see another example of dividing numbers written in scientific notation in the following video.\r\n\r\nhttps:\/\/youtu.be\/RlZck2W5pO4\r\n\r\nThe following video is a mini-lesson on how to convert decimals to scientific notation, and back to a decimal. Additionally, you will see more examples of how to multiply and divide numbers given in scientific notation.\r\n\r\nhttps:\/\/youtu.be\/hY-ecKyZ244\r\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply numbers expressed in\u00a0scientific notation<\/li>\n<li>Divide numbers expressed in scientific notation<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title2\">Multiplying Numbers Expressed in Scientific Notation<\/h2>\n<p>Numbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of the properties of numbers and the rules of exponents that you may recall. To multiply numbers in scientific notation, first multiply the numbers that aren\u2019t powers of 10 (the <i>a<\/i> in [latex]a\\times10^{n}[\/latex]). Then multiply the powers of ten by adding the exponents.<\/p>\n<p>This will produce a new number times a different power of [latex]10[\/latex]. All you have to do is check to make sure this new value is in scientific notation. If it isn\u2019t, you convert it.<\/p>\n<p>Let\u2019s look at some examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p style=\"text-align: center;\">[latex]\\left(3\\times10^{8}\\right)\\left(6.8\\times10^{-13}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q395606\">Show Solution<\/span><\/p>\n<div id=\"q395606\" class=\"hidden-answer\" style=\"display: none\">Regroup using the commutative and associative properties.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3\\times6.8\\right)\\left(10^{8}\\times10^{-13}\\right)[\/latex]<\/p>\n<p>Multiply the coefficients.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(20.4\\right)\\left(10^{8}\\times10^{-13}\\right)[\/latex]<\/p>\n<p>Multiply the powers of [latex]10[\/latex] using the Product Rule. Add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]20.4\\times10^{-5}[\/latex]<\/p>\n<p>Convert [latex]20.4[\/latex] into scientific notation by moving the decimal point one place to the left and multiplying by [latex]10^{1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2.04\\times10^{1}\\right)\\times10^{-5}[\/latex]<\/p>\n<p>Group the powers of [latex]10[\/latex] using the associative property of multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]2.04\\times\\left(10^{1}\\times10^{-5}\\right)[\/latex]<\/p>\n<p>Multiply using the Product Rule\u2014add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]2.04\\times10^{1+\\left(-5\\right)}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3\\times10^{8}\\right)\\left(6.8\\times10^{-13}\\right)=2.04\\times10^{-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p style=\"text-align: center;\">[latex]\\left(8.2\\times10^{6}\\right)\\left(1.5\\times10^{-3}\\right)\\left(1.9\\times10^{-7}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23947\">Show Solution<\/span><\/p>\n<div id=\"q23947\" class=\"hidden-answer\" style=\"display: none\">Regroup using the commutative and associative properties.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(8.2\\times1.5\\times1.9\\right)\\left(10^{6}\\times10^{-3}\\times10^{-7}\\right)[\/latex]<\/p>\n<p>Multiply the numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(23.37\\right)\\left(10^{6}\\times10^{-3}\\times10^{-7}\\right)[\/latex]<\/p>\n<p>Multiply the powers of [latex]10[\/latex] using the Product Rule\u2014add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]23.37\\times10^{-4}[\/latex]<\/p>\n<p>Convert [latex]23.37[\/latex] into scientific notation by moving the decimal point one place to the left and multiplying by [latex]10^{1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2.337\\times10^{1}\\right)\\times10^{-4}[\/latex]<\/p>\n<p>Group the powers of [latex]10[\/latex] using the associative property of multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]2.337\\times\\left(10^{1}\\times10^{-4}\\right)[\/latex]<\/p>\n<p>Multiply using the Product Rule and add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]2.337\\times10^{1+\\left(-4\\right)}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(8.2\\times10^{6}\\right)\\left(1.5\\times10^{-3}\\right)\\left(1.9\\times10^{-7}\\right)=2.337\\times10^{-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply. Write answers in decimal form: [latex]\\left(4\\times {10}^{5}\\right)\\left(2\\times {10}^{-7}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q630924\">Show Solution<\/span><\/p>\n<div id=\"q630924\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469641319\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(4\\times {10}^{5}\\right)\\left(2\\times {10}^{-7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to rearrange the factors.<\/td>\n<td>[latex]4\\cdot 2\\cdot {10}^{5}\\cdot {10}^{-7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]4[\/latex] by [latex]2[\/latex] and use the Product Property to multiply [latex]{10}^{5}[\/latex] by [latex]{10}^{-7}[\/latex].<\/td>\n<td>[latex]8\\times {10}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change to decimal form by moving the decimal two places left.<\/td>\n<td>[latex]{\\Large\\frac{8}{100}} = 0.08[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146318\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146318&theme=oea&iframe_resize_id=ohm146318&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2826\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2826&theme=oea&iframe_resize_id=ohm2826&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video you will see an example of how to multiply tow numbers that are written in scientific notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Examples:  Multiplying Numbers Written in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5ZAY4OCkp7U?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Dividing Numbers Expressed in Scientific Notation<\/h2>\n<p>In order to divide numbers in scientific notation, you once again apply the properties of numbers and the rules of exponents. You begin by dividing the numbers that aren\u2019t powers of [latex]10[\/latex] (the <i>a<\/i> in [latex]a\\times10^{n}[\/latex]. Then you divide the powers of ten by subtracting the exponents.<\/p>\n<p>This will produce a new number times a different power of 10. If it isn\u2019t already in scientific notation, you convert it, and then you\u2019re done.<\/p>\n<p>Let\u2019s look at some examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{2.829\\times 1{{0}^{-9}}}{3.45\\times 1{{0}^{-3}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q364796\">Show Solution<\/span><\/p>\n<div id=\"q364796\" class=\"hidden-answer\" style=\"display: none\">Regroup using the associative property.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{2.829}{3.45} \\right)\\left( \\frac{{{10}^{-9}}}{{{10}^{-3}}} \\right)[\/latex]<\/p>\n<p>Divide the coefficients.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left(0.82\\right)\\left( \\frac{{{10}^{-9}}}{{{10}^{-3}}} \\right)[\/latex]<\/p>\n<p>Divide the powers of [latex]10[\/latex] using the Quotient Rule. Subtract the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}0.82\\times10^{-9-\\left(-3\\right)}\\\\0.82\\times10^{-6}\\end{array}[\/latex]<\/p>\n<p>Convert [latex]0.82[\/latex] into scientific notation by moving the decimal point one place to the right and multiplying by [latex]10^{-1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(8.2\\times10^{-1}\\right)\\times10^{-6}[\/latex]<\/p>\n<p>Group the powers of [latex]10[\/latex] together using the associative property.<\/p>\n<p style=\"text-align: center;\">[latex]8.2\\times\\left(10^{-1}\\times10^{-6}\\right)[\/latex]<\/p>\n<p>Multiply the powers of [latex]10[\/latex] using the Product Rule\u2014add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]8.2\\times10^{-1+\\left(-6\\right)}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{2.829\\times {{10}^{-9}}}{3.45\\times {{10}^{-3}}}=8.2\\times {{10}^{-7}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{\\left(1.37\\times10^{4}\\right)\\left(9.85\\times10^{6}\\right)}{5.0\\times10^{12}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q337143\">Show Solution<\/span><\/p>\n<div id=\"q337143\" class=\"hidden-answer\" style=\"display: none\">Regroup the terms in the numerator according to the associative and commutative properties.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{\\left( 1.37\\times 9.85 \\right)\\left( {{10}^{6}}\\times {{10}^{4}} \\right)}{5.0\\times {{10}^{12}}}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{13.4945\\times {{10}^{10}}}{5.0\\times {{10}^{12}}}[\/latex]<\/p>\n<p>Regroup using the associative property.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{13.4945}{5.0} \\right)\\left( \\frac{{{10}^{10}}}{{{10}^{12}}} \\right)[\/latex]<\/p>\n<p>Divide the numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left(2.6989\\right)\\left(\\frac{10^{10}}{10^{12}}\\right)[\/latex]<\/p>\n<p>Divide the powers of [latex]10[\/latex] using the Quotient Rule\u2014subtract the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{c}\\left(2.6989 \\right)\\left( {{10}^{10-12}} \\right)\\\\2.6989\\times {{10}^{-2}}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{\\left( 1.37\\times {{10}^{4}} \\right)\\left( 9.85\\times {{10}^{6}} \\right)}{5.0\\times {{10}^{12}}}=2.6989\\times {{10}^{-2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Write answers in decimal form: [latex]{\\Large\\frac{9\\times {10}^{3}}{3\\times {10}^{-2}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q424217\">Show Solution<\/span><\/p>\n<div id=\"q424217\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469868577\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{9\\times {10}^{3}}{3\\times {10}^{-2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Separate the factors.<\/td>\n<td>[latex]{\\Large\\frac{9}{3}}\\times {\\Large\\frac{{10}^{3}}{{10}^{-2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]9[\/latex] by [latex]3[\/latex] and use the Quotient Property to divide [latex]{10}^{3}[\/latex] by [latex]{10}^{-2}[\/latex] .<\/td>\n<td>[latex]3\\times {10}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change to decimal form by moving the decimal five places right.<\/td>\n<td>[latex]300,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146319\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146319&theme=oea&iframe_resize_id=ohm146319&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"ohm2829\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2829&theme=oea&iframe_resize_id=ohm2829&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that when you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator. You will see another example of dividing numbers written in scientific notation in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Examples:  Dividing Numbers Written in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RlZck2W5pO4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following video is a mini-lesson on how to convert decimals to scientific notation, and back to a decimal. Additionally, you will see more examples of how to multiply and divide numbers given in scientific notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"(New Version Available)Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hY-ecKyZ244?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-16232\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples: Dividing Numbers Written in Scientific Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RlZck2W5pO4\">https:\/\/youtu.be\/RlZck2W5pO4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples: Multiplying Numbers Written in Scientific Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5ZAY4OCkp7U\">https:\/\/youtu.be\/5ZAY4OCkp7U<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":169554,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and 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