{"id":4428,"date":"2020-04-13T13:06:56","date_gmt":"2020-04-13T13:06:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4428"},"modified":"2021-02-06T00:05:06","modified_gmt":"2021-02-06T00:05:06","slug":"scientific-notation","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/scientific-notation\/","title":{"raw":"Scientific Notation","rendered":"Scientific Notation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Convert from decimal notation to scientific notation<\/li>\r\n \t<li>Convert from scientific notation to decimal notation<\/li>\r\n \t<li>Multiply and divide using scientific notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Convert from Decimal Notation to Scientific Notation<\/h3>\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\\times 1000[\/latex] and [latex]0.004[\/latex] means [latex]4\\times {\\Large\\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}{cccc}4000\\hfill &amp; &amp; &amp; 0.004\\hfill \\\\ 4\\times 1000\\hfill &amp; &amp; &amp; 4\\times {\\Large\\frac{1}{1000}}\\hfill \\\\ 4\\times {10}^{3}\\hfill &amp; &amp; &amp; 4\\times {\\Large\\frac{1}{{10}^{3}}}\\hfill \\\\ &amp; &amp; &amp; \\hfill 4\\times {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 <em>scientific notation.<\/em>\r\n<div class=\"textbox shaded\">\r\n<h3>Scientific Notation<\/h3>\r\nA number is expressed in scientific notation when it is of the form\r\n<p style=\"padding-left: 60px;\">[latex]a\\times {10}^{n}[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">where [latex]a\\ge 1[\/latex] and [latex]a&lt;10[\/latex] and [latex]n[\/latex] is an integer.<\/p>\r\n\r\n<\/div>\r\nIt is customary in scientific notation to use [latex]\\times [\/latex] as the multiplication sign, even though we avoid using this sign elsewhere in algebra.\r\n\r\nScientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.\r\n\r\nIf we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224558\/CNX_BMath_Figure_10_05_008_img.png\" 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, \" \/>\r\nIn both cases, the decimal was moved [latex]3[\/latex] places to get the first factor, [latex]4[\/latex], by itself.\r\n<ul id=\"fs-id1487398\">\r\n \t<li>The power of [latex]10[\/latex] is positive when the number is larger than [latex]1\\text{ : }4000=4\\times {10}^{3}[\/latex].<\/li>\r\n \t<li>The power of [latex]10[\/latex] is negative when the number is between [latex]0[\/latex] and [latex]1\\text{ : }0.004=4\\times {10}^{-3}[\/latex].<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite [latex]37,000[\/latex] in scientific notation.\r\n\r\nSolution\r\n<table id=\"eip-id1168467360549\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1<\/strong>: Move the decimal point so that the first factor is greater than or equal to [latex]1[\/latex] but less than [latex]10[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224600\/CNX_BMath_Figure_10_05_009_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2<\/strong>: Count the number of decimal places, [latex]n[\/latex] , that the decimal point was moved.<\/td>\r\n<td>[latex]3.70000[\/latex]\r\n\r\n[latex]4[\/latex] places<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3<\/strong>: Write the number as a product with a power of [latex]10[\/latex].<\/td>\r\n<td>[latex]3.7\\times {10}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If the original number is:\r\n<ul id=\"eip-id1168468452809\">\r\n \t<li>greater than 1, the power of 10 will be [latex]{10}^{n}[\/latex] .<\/li>\r\n \t<li>between 0 and 1, the power of 10 will be [latex]{10}^{\\mathrm{-n}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 4<\/strong>: Check.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{10}^{4}[\/latex] is [latex]10,000[\/latex] and [latex]10,000[\/latex] times [latex]3.7[\/latex] will be [latex]37,000[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]37,000=3.7\\times {10}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146311[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Convert from decimal notation to scientific notation<\/h3>\r\n<ol id=\"eip-id1168468758847\" class=\"stepwise\">\r\n \t<li>Move the decimal point so that the first factor is greater than or equal to [latex]1[\/latex] but less than [latex]10[\/latex].<\/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-id1168466026987\">\r\n \t<li>If the original number is:\r\n<ul id=\"eip-id4925542\">\r\n \t<li>greater than [latex]1[\/latex], the power of [latex]10[\/latex] will be [latex]{10}^{n}[\/latex].<\/li>\r\n \t<li>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<\/ul>\r\n<\/li>\r\n \t<li>Check.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite in scientific notation: [latex]0.0052[\/latex]\r\n[reveal-answer q=\"861049\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"861049\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467187199\" class=\"unnumbered unstyled\" summary=\"The top line shows 0.0052. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.0052[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Move the decimal point to get [latex]5.2[\/latex], a number between [latex]1[\/latex] and [latex]10[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224601\/CNX_BMath_Figure_10_05_010_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of decimal places the point was moved.<\/td>\r\n<td>[latex]3[\/latex] places<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a product with a power of [latex]10[\/latex].<\/td>\r\n<td>[latex]5.2\\times {10}^{-3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer:\r\n\r\n[latex]\\begin{array}{c}\\hfill 5.2\\times {10}^{-3}\\hfill \\\\ \\hfill 5.2\\times {\\Large\\frac{1}{{10}^{3}}}\\hfill \\\\ \\\\ \\\\ \\hfill 5.2\\times {\\Large\\frac{1}{1000}}\\hfill \\\\ \\hfill 5.2\\times 0.001\\hfill \\\\ \\hfill 0.0052\\hfill \\end{array}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.0052=5.2\\times {10}^{-3}[\/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]146312[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Convert Scientific Notation to Decimal Form<\/h3>\r\nHow can we convert from scientific notation to decimal form? Let\u2019s look at two numbers written in scientific notation and see.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}9.12\\times {10}^{4}\\hfill &amp; &amp; &amp; 9.12\\times {10}^{-4}\\hfill \\\\ 9.12\\times 10,000\\hfill &amp; &amp; &amp; 9.12\\times 0.0001\\hfill \\\\ 91,200\\hfill &amp; &amp; &amp; 0.000912\\hfill \\end{array}[\/latex]<\/p>\r\nIf 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224602\/CNX_BMath_Figure_10_05_011_img.png\" 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.\" \/>\r\nIn both cases the decimal point moved 4 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<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nConvert to decimal form: [latex]6.2\\times {10}^{3}[\/latex]\r\n[reveal-answer q=\"593540\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"593540\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168465244409\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1<\/strong>: Determine the exponent, [latex]n[\/latex] , on the factor [latex]10[\/latex].<\/td>\r\n<td>[latex]6.2\\times {10}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2<\/strong>: Move the decimal point [latex]n[\/latex] places, adding zeros if needed.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224604\/CNX_BMath_Figure_10_05_012_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<ul id=\"eip-id1168467377733\">\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<\/td>\r\n<td>[latex]6,200[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3<\/strong>: Check to see if your answer makes sense.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{10}^{3}[\/latex] is [latex]1000[\/latex] and [latex]1000[\/latex] times [latex]6.2[\/latex] will be [latex]6,200[\/latex].<\/td>\r\n<td>[latex]6.2\\times {10}^{3}=6,200[\/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]146313[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Convert scientific notation to decimal form<\/h3>\r\n<ol id=\"eip-id1168469761724\" 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-id1168466070183\">\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<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nConvert to decimal form: [latex]8.9\\times {10}^{-2}[\/latex]\r\n[reveal-answer q=\"278675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"278675\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469657168\" class=\"unnumbered unstyled\" summary=\"The top line shows 8.9 times 10 to the negative 2. The next line says, \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]8.9\\times {10}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Determine the exponent [latex]n[\/latex] , on the factor [latex]10[\/latex].<\/td>\r\n<td style=\"height: 15px;\">The exponent is [latex]\u22122[\/latex].<\/td>\r\n<\/tr>\r\n<tr style=\"height: 33px;\">\r\n<td style=\"height: 33px;\">Move the decimal point [latex]2[\/latex] places to the left.<\/td>\r\n<td style=\"height: 33px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224605\/CNX_BMath_Figure_10_05_013_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.3125px;\">\r\n<td style=\"height: 15.3125px;\">Add zeros as needed for placeholders.<\/td>\r\n<td style=\"height: 15.3125px;\">[latex]0.089[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]8.9\\times {10}^{-2}=0.089[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">The Check is left to you.<\/td>\r\n<td style=\"height: 15px;\"><\/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]146314[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Multiply and Divide Using Scientific Notation<\/h3>\r\nWe use the Properties of Exponents to multiply and divide numbers in scientific notation.\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<\/div>\r\n&nbsp;\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\r\n<\/div>\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","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Convert from decimal notation to scientific notation<\/li>\n<li>Convert from scientific notation to decimal notation<\/li>\n<li>Multiply and divide using scientific notation<\/li>\n<\/ul>\n<\/div>\n<h3>Convert from Decimal Notation to Scientific Notation<\/h3>\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\\times 1000[\/latex] and [latex]0.004[\/latex] means [latex]4\\times {\\Large\\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}{cccc}4000\\hfill & & & 0.004\\hfill \\\\ 4\\times 1000\\hfill & & & 4\\times {\\Large\\frac{1}{1000}}\\hfill \\\\ 4\\times {10}^{3}\\hfill & & & 4\\times {\\Large\\frac{1}{{10}^{3}}}\\hfill \\\\ & & & \\hfill 4\\times {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 <em>scientific notation.<\/em><\/p>\n<div class=\"textbox shaded\">\n<h3>Scientific Notation<\/h3>\n<p>A number is expressed in scientific notation when it is of the form<\/p>\n<p style=\"padding-left: 60px;\">[latex]a\\times {10}^{n}[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">where [latex]a\\ge 1[\/latex] and [latex]a<10[\/latex] and [latex]n[\/latex] is an integer.<\/p>\n<\/div>\n<p>It is customary in scientific notation to use [latex]\\times[\/latex] as the multiplication sign, even though we avoid using this sign elsewhere in algebra.<\/p>\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.<\/p>\n<p>If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224558\/CNX_BMath_Figure_10_05_008_img.png\" 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,\" \/><br \/>\nIn both cases, the decimal was moved [latex]3[\/latex] places to get the first factor, [latex]4[\/latex], by itself.<\/p>\n<ul id=\"fs-id1487398\">\n<li>The power of [latex]10[\/latex] is positive when the number is larger than [latex]1\\text{ : }4000=4\\times {10}^{3}[\/latex].<\/li>\n<li>The power of [latex]10[\/latex] is negative when the number is between [latex]0[\/latex] and [latex]1\\text{ : }0.004=4\\times {10}^{-3}[\/latex].<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write [latex]37,000[\/latex] in scientific notation.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467360549\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><strong>Step 1<\/strong>: Move the decimal point so that the first factor is greater than or equal to [latex]1[\/latex] but less than [latex]10[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224600\/CNX_BMath_Figure_10_05_009_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2<\/strong>: Count the number of decimal places, [latex]n[\/latex] , that the decimal point was moved.<\/td>\n<td>[latex]3.70000[\/latex]<\/p>\n<p>[latex]4[\/latex] places<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3<\/strong>: Write the number as a product with a power of [latex]10[\/latex].<\/td>\n<td>[latex]3.7\\times {10}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>If the original number is:<\/p>\n<ul id=\"eip-id1168468452809\">\n<li>greater than 1, the power of 10 will be [latex]{10}^{n}[\/latex] .<\/li>\n<li>between 0 and 1, the power of 10 will be [latex]{10}^{\\mathrm{-n}}[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 4<\/strong>: Check.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]{10}^{4}[\/latex] is [latex]10,000[\/latex] and [latex]10,000[\/latex] times [latex]3.7[\/latex] will be [latex]37,000[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]37,000=3.7\\times {10}^{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146311\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146311&theme=oea&iframe_resize_id=ohm146311&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Convert from decimal notation to scientific notation<\/h3>\n<ol id=\"eip-id1168468758847\" class=\"stepwise\">\n<li>Move the decimal point so that the first factor is greater than or equal to [latex]1[\/latex] but less than [latex]10[\/latex].<\/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-id1168466026987\">\n<li>If the original number is:\n<ul id=\"eip-id4925542\">\n<li>greater than [latex]1[\/latex], the power of [latex]10[\/latex] will be [latex]{10}^{n}[\/latex].<\/li>\n<li>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<\/ul>\n<\/li>\n<li>Check.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write in scientific notation: [latex]0.0052[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q861049\">Show Solution<\/span><\/p>\n<div id=\"q861049\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467187199\" class=\"unnumbered unstyled\" summary=\"The top line shows 0.0052. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.0052[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Move the decimal point to get [latex]5.2[\/latex], a number between [latex]1[\/latex] and [latex]10[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224601\/CNX_BMath_Figure_10_05_010_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of decimal places the point was moved.<\/td>\n<td>[latex]3[\/latex] places<\/td>\n<\/tr>\n<tr>\n<td>Write as a product with a power of [latex]10[\/latex].<\/td>\n<td>[latex]5.2\\times {10}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer:<\/p>\n<p>[latex]\\begin{array}{c}\\hfill 5.2\\times {10}^{-3}\\hfill \\\\ \\hfill 5.2\\times {\\Large\\frac{1}{{10}^{3}}}\\hfill \\\\ \\\\ \\\\ \\hfill 5.2\\times {\\Large\\frac{1}{1000}}\\hfill \\\\ \\hfill 5.2\\times 0.001\\hfill \\\\ \\hfill 0.0052\\hfill \\end{array}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0.0052=5.2\\times {10}^{-3}[\/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=\"ohm146312\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146312&theme=oea&iframe_resize_id=ohm146312&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Convert Scientific Notation to Decimal Form<\/h3>\n<p>How can we convert from scientific notation to decimal form? Let\u2019s look at two numbers written in scientific notation and see.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}9.12\\times {10}^{4}\\hfill & & & 9.12\\times {10}^{-4}\\hfill \\\\ 9.12\\times 10,000\\hfill & & & 9.12\\times 0.0001\\hfill \\\\ 91,200\\hfill & & & 0.000912\\hfill \\end{array}[\/latex]<\/p>\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.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224602\/CNX_BMath_Figure_10_05_011_img.png\" 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.\" \/><br \/>\nIn both cases the decimal point moved 4 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<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Convert to decimal form: [latex]6.2\\times {10}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q593540\">Show Solution<\/span><\/p>\n<div id=\"q593540\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168465244409\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><strong>Step 1<\/strong>: Determine the exponent, [latex]n[\/latex] , on the factor [latex]10[\/latex].<\/td>\n<td>[latex]6.2\\times {10}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2<\/strong>: Move the decimal point [latex]n[\/latex] places, adding zeros if needed.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224604\/CNX_BMath_Figure_10_05_012_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<ul id=\"eip-id1168467377733\">\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<\/td>\n<td>[latex]6,200[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3<\/strong>: Check to see if your answer makes sense.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]{10}^{3}[\/latex] is [latex]1000[\/latex] and [latex]1000[\/latex] times [latex]6.2[\/latex] will be [latex]6,200[\/latex].<\/td>\n<td>[latex]6.2\\times {10}^{3}=6,200[\/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=\"ohm146313\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146313&theme=oea&iframe_resize_id=ohm146313&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Convert scientific notation to decimal form<\/h3>\n<ol id=\"eip-id1168469761724\" 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-id1168466070183\">\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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Convert to decimal form: [latex]8.9\\times {10}^{-2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278675\">Show Solution<\/span><\/p>\n<div id=\"q278675\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469657168\" class=\"unnumbered unstyled\" summary=\"The top line shows 8.9 times 10 to the negative 2. The next line says,\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]8.9\\times {10}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Determine the exponent [latex]n[\/latex] , on the factor [latex]10[\/latex].<\/td>\n<td style=\"height: 15px;\">The exponent is [latex]\u22122[\/latex].<\/td>\n<\/tr>\n<tr style=\"height: 33px;\">\n<td style=\"height: 33px;\">Move the decimal point [latex]2[\/latex] places to the left.<\/td>\n<td style=\"height: 33px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224605\/CNX_BMath_Figure_10_05_013_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15.3125px;\">\n<td style=\"height: 15.3125px;\">Add zeros as needed for placeholders.<\/td>\n<td style=\"height: 15.3125px;\">[latex]0.089[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]8.9\\times {10}^{-2}=0.089[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">The Check is left to you.<\/td>\n<td style=\"height: 15px;\"><\/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=\"ohm146314\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146314&theme=oea&iframe_resize_id=ohm146314&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Multiply and Divide Using Scientific Notation<\/h3>\n<p>We use the Properties of Exponents to multiply and divide numbers in scientific notation.<\/p>\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<\/div>\n<p>&nbsp;<\/p>\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><\/p>\n<\/div>\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-1\" 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-4428\">\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>Question ID 146319, 146318, 146314, 146313, 146312, 146311. <strong>Authored 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>Scientific Notation. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hY-ecKyZ244\">https:\/\/youtu.be\/hY-ecKyZ244<\/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 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":25777,"menu_order":8,"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\"},{\"type\":\"original\",\"description\":\"Question ID 146319, 146318, 146314, 146313, 146312, 146311\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Scientific Notation\",\"author\":\"James Sousa 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