{"id":4333,"date":"2016-05-24T22:53:19","date_gmt":"2016-05-24T22:53:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nrocarithmetic\/?post_type=chapter&#038;p=4333"},"modified":"2018-01-03T23:57:41","modified_gmt":"2018-01-03T23:57:41","slug":"outcome-writing-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/beginalgebra\/chapter\/outcome-writing-scientific-notation\/","title":{"raw":"Scientific Notation","rendered":"Scientific Notation"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define decimal and scientific notation<\/li>\r\n \t<li>Convert between scientific and decimal notation<\/li>\r\n \t<li>Multiply and divide numbers expressed in\u00a0scientific notation<\/li>\r\n \t<li>Solve application problems involving scientific notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title1\">Convert between scientific and decimal notation<\/h2>\r\nBefore we can convert between scientific and decimal notation, we need to know the difference between the two.\u00a0<strong>S<\/strong><b>cientific notation <\/b>is used by\u00a0scientists, mathematicians, and engineers when they are working with very large or very small numbers.\u00a0Using exponential notation, large and small numbers can be written in a way that is easier to read.\r\n\r\nWhen a number is written in scientific notation, the <b>exponent<\/b> tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think\u00a0about it.\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>Word<\/td>\r\n<td>How many thousands<\/td>\r\n<td>Number<\/td>\r\n<td>Scientific Notation<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>million<\/td>\r\n<td>1000 x 1000 = a thousand thousands<\/td>\r\n<td>1,000,000<\/td>\r\n<td>\u00a0[latex]10^6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>billion<\/td>\r\n<td>(1000 x 1000) x 1000 = a thousand millions<\/td>\r\n<td>1,000,000,000<\/td>\r\n<td>\u00a0\u00a0[latex]10^9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>trillion<\/td>\r\n<td>(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\r\n<td>\u00a01,000,000,000,000<\/td>\r\n<td>\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n1 billion can be written as 1,000,000,000 or represented as\u00a0[latex]10^9[\/latex]. How would 2 billion be\u00a0represented? Since 2 billion is 2 times 1\u00a0billion, then 2 billion can be written as [latex]2\\times10^9[\/latex].\r\n\r\nA\u00a0light year is the number of miles light travels in one year, about 5,880,000,000,000. \u00a0That's a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number.\u00a0Using scientific notation, the distance is [latex]5.88\\times10^{12}[\/latex]\u00a0miles. The exponent of 12 tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a\u00a0hydrogen atom, which is about 0.00000005 mm, and in scientific notation is\u00a0[latex]5\\times10^{-8}[\/latex]\u00a0mm. In this case the [latex]-8[\/latex] tells us how many places to count to the right of the decimal.\r\n\r\nOutlined in the box below are some important conventions of scientific notation format.\r\n<div class=\"textbox shaded\">\r\n<h3>Scientific Notation<\/h3>\r\nA positive number is written in scientific notation if it is written as [latex]a\\times10^{n}[\/latex]\u00a0where the coefficient <i>a<\/i>\u00a0is [latex]1\\leq{a}&lt;10[\/latex], and <i>n <\/i>is an integer.\r\n\r\n<\/div>\r\nLook at the numbers below. Which of the numbers is written in scientific notation?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><b>Number<\/b><\/td>\r\n<td><b>Scientific Notation?<\/b><\/td>\r\n<td><b>Explanation<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1.85\\times10^{-2}[\/latex]<\/td>\r\n<td>yes<\/td>\r\n<td>[latex]1\\leq1.85&lt;10[\/latex]\r\n\r\n[latex]-2[\/latex] is an integer<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle 1.083\\times {{10}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>no<\/td>\r\n<td>[latex] \\displaystyle \\frac{1}{2}[\/latex] is not an integer<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0.82\\times10^{14}[\/latex]<\/td>\r\n<td>no<\/td>\r\n<td>0.82 is not [latex]\\geq1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10\\times10^{3}[\/latex]<\/td>\r\n<td>no<\/td>\r\n<td>10 is not &lt; 10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow let\u2019s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">\r\n<p style=\"text-align: center;\"><b>Large Numbers<\/b><\/p>\r\n<\/td>\r\n<td><b>\u00a0<\/b><\/td>\r\n<td colspan=\"2\">\r\n<p style=\"text-align: center;\"><b>Small Numbers<\/b><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><b>Decimal Notation<\/b><\/td>\r\n<td><b>Scientific Notation<\/b><\/td>\r\n<td><b>\u00a0<\/b><\/td>\r\n<td><b>Decimal Notation<\/b><\/td>\r\n<td><b>Scientific Notation<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>500.0<\/td>\r\n<td>[latex]5\\times10^{2}[\/latex]<\/td>\r\n<td><\/td>\r\n<td>0.05<\/td>\r\n<td>[latex]5\\times10^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>80,000.0<\/td>\r\n<td>[latex]8\\times10^{4}[\/latex]<\/td>\r\n<td><\/td>\r\n<td>0.0008<\/td>\r\n<td>[latex]8\\times10^{-4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>43,000,000.0<\/td>\r\n<td>[latex]4.3\\times10^{7}[\/latex]<\/td>\r\n<td><\/td>\r\n<td>0.00000043<\/td>\r\n<td>[latex]4.3\\times10^{-7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>62,500,000,000.0<\/td>\r\n<td>[latex]6.25\\times10^{10}[\/latex]<\/td>\r\n<td><\/td>\r\n<td>0.000000000625<\/td>\r\n<td>[latex]6.25\\times10^{-10}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<h3>Convert from decimal\u00a0notation to scientific\u00a0notation<\/h3>\r\nTo write a <i>large<\/i> number in scientific notation, move the decimal point to the left to obtain a number between 1 and 10. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 10 so that the expression has the same value.\r\n\r\nLet\u2019s look at an example.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}180,000.=18,000.0\\times10^{1}\\\\1,800.00\\times10^{2}\\\\180.000\\times10^{3}\\\\18.0000\\times10^{4}\\\\1.80000\\times10^{5}\\\\180,000=1.8\\times10^{5}\\end{array}[\/latex]<\/p>\r\nNotice that the decimal point was moved 5 places to the left, and the exponent is 5.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following numbers in scientific notation.\r\n<ol>\r\n \t<li>[latex]920,000,000[\/latex]<\/li>\r\n \t<li>[latex]10,200,000[\/latex]<\/li>\r\n \t<li>[latex]100,000,000,000[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"628\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"628\"]\r\n<ol>\r\n \t<li>[latex]\\underset{\\longleftarrow}{920,000,000}[\/latex] \u00a0We will move the decimal point to the left, it helps to place it\u00a0at the end of the number and then count how many times you move it to get one number before it that is between 1 and 10.\u00a0[latex]\\underset{\\longleftarrow}{920,000,000}=920,000,000.0[\/latex], move the decimal point 8 times to the left and you will have\u00a0[latex]9.20,000,000[\/latex], now we can replace the zeros with an exponent of 8,\u00a0[latex]9.2\\times10^{8}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{\\longleftarrow}{10,200,000}=10,200,000.0=1.02\\times10^{7}[\/latex], note here how we included the 0 and the 2 after the decimal point. \u00a0In some disciplines, you may learn about when to include both of these. \u00a0Follow instructions from your teacher on rounding rules.<\/li>\r\n \t<li>[latex]\\underset{\\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\\times10^{11}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nTo write a small number (between 0 and 1) in scientific notation, you move the decimal to the <i>right<\/i> and the exponent will have to be <i>negative,\u00a0<\/i>as in the following example.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underset{\\longrightarrow}{0.00004}=00.0004\\times10^{-1}\\\\000.004\\times10^{-2}\\\\0000.04\\times10^{-3}\\\\00000.4\\times10^{-4}\\\\000004.\\times10^{-5}\\\\0.00004=4\\times10^{-5}\\end{array}[\/latex]<\/p>\r\nYou may notice that the decimal point was moved five places to the <i>right <\/i>until you got \u00a0to the number 4, which is between 1 and 10. The exponent is [latex]\u22125[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following numbers in scientific notation.\r\n<ol>\r\n \t<li>[latex]0.0000000000035[\/latex]<\/li>\r\n \t<li>[latex]0.0000000102[\/latex]<\/li>\r\n \t<li>[latex]0.00000000000000793[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"229054\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229054\"]\r\n<ol>\r\n \t<li>[latex]\\underset{\\longrightarrow}{0.0000000000035}=3.5\\times10^{-12}[\/latex], we moved the decimal 12 times to get to a number between 1 and 10<\/li>\r\n \t<li>[latex]\\underset{\\longrightarrow}{0.0000000102}=1.02\\times10^{-8}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{\\longrightarrow}{0.00000000000000793}=7.93\\times10^{-15}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation.\r\n\r\nhttps:\/\/youtu.be\/fsNu3AdIgdk\r\n<h3>Convert from scientific notation to decimal notation<\/h3>\r\nYou can also write scientific notation as decimal notation. Recall\u00a0the number of miles that light travels in a year is [latex]5.88\\times10^{12}[\/latex], and a hydrogen atom has a diameter of [latex]5\\times10^{-8}[\/latex]\u00a0mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is<i> positive<\/i>, move the decimal point to the <i>right.<\/i> If the exponent is<i> negative<\/i>, move the decimal point to the <i>left.<\/i>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5.88\\times10^{12}=\\underset{\\longrightarrow}{5.880000000000.}=5,880,000,000,000\\\\5\\times10^{-8}=\\underset{\\longleftarrow}{0.00000005.}=0.00000005\\end{array}[\/latex]<\/p>\r\nFor each power of 10, you move the decimal point one place. Be careful here and don\u2019t get carried away with the zeros\u2014the number of zeros after the decimal point will always be 1 <i>less<\/i> than the exponent because it takes one power of 10 to shift that first number to the left of the decimal.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following in decimal notation.\r\n<ol>\r\n \t<li>[latex]4.8\\times10{-4}[\/latex]<\/li>\r\n \t<li>[latex]3.08\\times10^{6}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"489774\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489774\"]\r\n<ol>\r\n \t<li>[latex]4.8\\times10^{-4}[\/latex], the exponent is negative, so we need to move the decimal to the left. \u00a0[latex]\\underset{\\longleftarrow}{4.8\\times10^{-4}}=\\underset{\\longleftarrow}{.00048}[\/latex]<\/li>\r\n \t<li>[latex]3.08\\times10^{6}[\/latex], the exponent is positive, so we need to move the decimal to the right. \u00a0[latex]\\underset{\\longrightarrow}{3.08\\times10^{6}}=\\underset{\\longrightarrow}{3080000}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nTo help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to wirte your ideas before you reveal the solution.\r\n\r\n1. You are writing a number that is greater than 1 in scientific notation. \u00a0Will your exponent be positive or negative?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n2. You are writing a number that is between 0 and 1 in scientific notation. \u00a0Will your exponent be positive or negative?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n3. What power do you need to put on 10 to get a result of 1?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"824936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"824936\"]\r\n1. <em>You are writing a number that is greater than 1 in scientific notation. Will your exponent be positive or negative?<\/em> For numbers greater than 1, the exponent on 10 will be positive when you are using scientific notation. Refer to the table presented above:\r\n<table class=\" undefined\">\r\n<thead>\r\n<tr>\r\n<td>Word<\/td>\r\n<td>How many thousands<\/td>\r\n<td>Number<\/td>\r\n<td>Scientific Notation<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>million<\/td>\r\n<td>1000 x 1000 = a thousand thousands<\/td>\r\n<td>1,000,000<\/td>\r\n<td>\u00a0[latex]10^6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>billion<\/td>\r\n<td>(1000 x 1000) x 1000 = a thousand millions<\/td>\r\n<td>1,000,000,000<\/td>\r\n<td>\u00a0\u00a0[latex]10^9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>trillion<\/td>\r\n<td>(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\r\n<td>\u00a01,000,000,000,000<\/td>\r\n<td>\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. <em>You are writing a number that is between 0 and 1 in scientific notation. Will your exponent be positive or negative?<\/em> We can reason that since numbers greater than 1 will have a positive exponent, numbers between\u00a00 and 1 will have a negative exponent. Why are we specifying numbers between 0 and 1? The numbers between 0 and 1 represent amounts that are fractional. Recall that we defined numbers with a negative exponent as\u00a0[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex], so if we have [latex]10^{-2}[\/latex] we have [latex]\\frac{1}{10\\times10}=\\frac{1}{100}[\/latex] which is a number between 0 and 1.\r\n\r\n3.\u00a0<em>What power do you need to put on 10 to get a result of 1?<\/em> Recall\u00a0that any number or variable with an exponent of 0 is equal to 1, as in this example:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1\\\\\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}\\\\\\text{ therefore }\\\\{t}^{0}=1\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We now have described the notation necessary to write all possible numbers on the number line in scientific notation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the next video you will see how to convert a number written in scientific notation into decimal notation.\r\n\r\nhttps:\/\/youtu.be\/8BX0oKUMIjw\r\n<h2 id=\"title2\">Multiplying and Dividing 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 10. 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 10 using the Product Rule. Add the exponents.\r\n<p style=\"text-align: center;\">[latex]20.4\\times10^{-5}[\/latex]<\/p>\r\nConvert 20.4 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 10 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 10 using the Product Rule\u2014add the exponents.\r\n<p style=\"text-align: center;\">[latex]23.37\\times10^{-4}[\/latex]<\/p>\r\nConvert 23.37 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 10 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\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\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 10 (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 10 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 0.82 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 10 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 10 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 10 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\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<h2 id=\"title3\">Solve application problems<\/h2>\r\n[caption id=\"attachment_4384\" align=\"aligncenter\" width=\"361\"]<img class=\"wp-image-4384\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26225033\/Screen-Shot-2016-05-26-at-3.50.25-PM-300x197.png\" alt=\"Molecule of water with one oxygen bonded to two hydrogen.\" width=\"361\" height=\"237\" \/> Water Molecule[\/caption]\r\n\r\nLearning rules for exponents\u00a0seems pointless without context, so let's explore some examples of using scientific notation that involve real problems. First, let's look at an example of how scientific notation can be used to describe real measurements.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nMatch each length in the table with the appropriate number of meters described in scientific notation below.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>The height of a desk<\/td>\r\n<td>Diameter of\u00a0water molecule<\/td>\r\n<td>Diameter of Sun at its equator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distance from Earth to Neptune<\/td>\r\n<td>Diameter of Earth at the Equator<\/td>\r\n<td>Height of Mt. Everest (rounded)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Diameter\u00a0of average human cell<\/td>\r\n<td>Diameter of a large grain of sand<\/td>\r\n<td>Distance a bullet travels in one second<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>Power of 10, units in meters<\/td>\r\n<td>Length from table above<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]10^{12}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{9}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{6}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{4}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{2}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{0}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{-3}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{-5}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{-10}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"993302\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"993302\"]\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>Power of 10, units in meters<\/td>\r\n<td>Length from table above<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]10^{12}[\/latex]<\/td>\r\n<td>Distance from Earth to Neptune<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{9}[\/latex]<\/td>\r\n<td>Diameter of Sun at it's Equator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{6}[\/latex]<\/td>\r\n<td>Diameter of Earth at the Equator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{4}[\/latex]<\/td>\r\n<td>Height of Mt. Everest (rounded)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{2}[\/latex]<\/td>\r\n<td>Distance a bullet travels in one second<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{0}[\/latex]<\/td>\r\n<td>The height of a desk<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{-3}[\/latex]<\/td>\r\n<td>Diameter of a large grain of sand<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{-5}[\/latex]<\/td>\r\n<td>Diameter\u00a0of average human cell<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10^{-10}[\/latex]<\/td>\r\n<td>Diameter of\u00a0water molecule<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption id=\"attachment_4382\" align=\"alignright\" width=\"181\"]<img class=\" wp-image-4382\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26224141\/Screen-Shot-2016-05-26-at-3.41.43-PM-300x210.png\" alt=\"Red Blood Cells.\" width=\"181\" height=\"126\" \/> Red Blood Cells[\/caption]\r\n\r\nOne of the most important parts of solving a \"real\" problem is translating the words into appropriate mathematical terms, and recognizing when a well known formula may help.\u00a0Here's an example that requires you to find the density\u00a0of a cell, given its mass and volume. Cells aren't visible to the naked eye, so their measurements, as described with scientific notation, involve negative exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nHuman cells come in a wide variety of shapes and sizes. The mass of an average human cell is about [latex]2\\times10^{-11}[\/latex] grams[footnote]Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https:\/\/en.wikipedia.org\/wiki\/Orders_of_magnitude_(mass)[\/footnote]Red blood cells are one of the smallest types of cells[footnote]How Big is a Human Cell?[\/footnote], clocking in at a volume of approximately [latex]10^{-6}\\text{ meters }^3[\/latex].[footnote]How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from http:\/\/www.weizmann.ac.il\/plants\/Milo\/images\/humanCellSize120116Clean.pdf[\/footnote] Biologists have recently discovered how to use the density of some types of cells to indicate the presence of disorders such as sickle cell anemia or leukemia. [footnote]Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., &amp; Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073\/pnas.1104651108[\/footnote]\u00a0Density is calculated as the ratio of [latex]\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]. Calculate the density of an average human cell.\r\n\r\n[reveal-answer q=\"856454\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"856454\"]\r\n\r\n<strong>Read and Understand:\u00a0<\/strong>We are given an average cellular mass and volume as well as the formula for density. We are looking for the density of an average human cell.\r\n\r\n<strong>Define and Translate:\u00a0<\/strong>\u00a0[latex]m=\\text{mass}=2\\times10^{-11}[\/latex], [latex]v=\\text{volume}=10^{-6}\\text{ meters}^3[\/latex], [latex]\\text{density}=\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]\r\n\r\n<strong>Write and Solve:\u00a0<\/strong>Use the quotient rule to simplify the ratio.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{ density }=\\frac{2\\times10^{-11}\\text{ grams }}{10^{-6}\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-11-\\left(-6\\right)}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\end{array}[\/latex]<\/p>\r\nIf scientists know the density of healthy cells, they can compare the density of a sick person's\u00a0cells to that to rule out or test for disorders or diseases that may affect cellular density.\r\n<h4 style=\"text-align: left;\">Answer<\/h4>\r\n<p style=\"text-align: left;\">The average density of a human cell is [latex]2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nThe following video provides an example of how to find the number of operations a computer can perform in a very short amount of time.\r\n\r\nhttps:\/\/youtu.be\/Cbm6ejEbu-o\r\n\r\n[caption id=\"attachment_4381\" align=\"alignright\" width=\"342\"]<img class=\" wp-image-4381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26223918\/Screen-Shot-2016-05-26-at-3.39.02-PM-300x137.png\" alt=\"Earth in the foreground, sun in the background, light beams traveling from the sun to the earth.\" width=\"342\" height=\"156\" \/> Light traveling from the sun to the earth.[\/caption]\r\n\r\nIn the next example, you will use another well known formula, [latex]d=r\\cdot{t}[\/latex], to find how long it takes light to travel from the sun to the earth. Unlike the previous example, the distance between the earth and the sun is massive, so the numbers you will work with have positive exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe speed of light is [latex]3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}[\/latex]. If the sun is [latex]1.5\\times10^{11}[\/latex] meters from earth, how many seconds does it take for sunlight to reach the earth? \u00a0Write your answer in scientific notation.\r\n[reveal-answer q=\"532092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"532092\"]\r\n\r\n<strong>Read and Understand:\u00a0<\/strong>We are looking for how long\u2014an amount of time. We are given a rate which has units of meters per second and a distance in meters. This is a [latex]d=r\\cdot{t}[\/latex] problem.\r\n\r\n<strong>Define and Translate:\u00a0<\/strong>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=1.5\\times10^{11}\\\\r=3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}\\\\t=\\text{ ? }\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\"><strong>Write and Solve:\u00a0<\/strong>Substitute the values we are given into the\u00a0[latex]d=r\\cdot{t}[\/latex] equation. We will work without units to make it easier. Often, scientists will work with units to make sure they have made correct calculations.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}d=r\\cdot{t}\\\\1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Divide both sides of the equation by\u00a0[latex]3\\times10^{8}[\/latex] to isolate\u00a0<em>t.<\/em><\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\\\\\text{ }\\\\\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with\u00a0<em>t.\u00a0<\/em><\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\\\\\text{ }\\\\\\left(\\frac{1.5}{3}\\right)\\times\\left(\\frac{10^{11}}{10^{8}}\\right)=t\\\\\\text{ }\\\\\\left(0.5\\right)\\times\\left(10^{11-8}\\right)=t\\\\0.5\\times10^3=t\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">This answer is not in scientific notation, so we will move the decimal to the right, which means we need to subtract one factor of 10.<\/p>\r\n<p style=\"text-align: center;\">[latex]0.5\\times10^3=5.0\\times10^2=t[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe time it takes light to travel from the sun to the earth is [latex]5.0\\times10^2=t[\/latex] seconds, or in standard\u00a0notation, 500 seconds. \u00a0That's not bad considering how far it has to travel!\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video\u00a0we calculate how many miles the participants of the New York marathon ran combined, and compare that to the circumference of the earth.\r\nhttps:\/\/youtu.be\/15tw4-v100Y\r\n\r\n&nbsp;\r\n<h3>Summary<\/h3>\r\nLarge and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.\r\n\r\nScientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of 10. The format is written [latex]a\\times10^{n}[\/latex], where [latex]1\\leq{a}&lt;10[\/latex]\u00a0and <i>n <\/i>is an integer.\u00a0To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.\r\n\r\n&nbsp;\r\n<h2 id=\"video1\" class=\"no-indent\" style=\"text-align: left;\"><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define decimal and scientific notation<\/li>\n<li>Convert between scientific and decimal notation<\/li>\n<li>Multiply and divide numbers expressed in\u00a0scientific notation<\/li>\n<li>Solve application problems involving scientific notation<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title1\">Convert between scientific and decimal notation<\/h2>\n<p>Before we can convert between scientific and decimal notation, we need to know the difference between the two.\u00a0<strong>S<\/strong><b>cientific notation <\/b>is used by\u00a0scientists, mathematicians, and engineers when they are working with very large or very small numbers.\u00a0Using exponential notation, large and small numbers can be written in a way that is easier to read.<\/p>\n<p>When a number is written in scientific notation, the <b>exponent<\/b> tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think\u00a0about it.<\/p>\n<table>\n<thead>\n<tr>\n<td>Word<\/td>\n<td>How many thousands<\/td>\n<td>Number<\/td>\n<td>Scientific Notation<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>million<\/td>\n<td>1000 x 1000 = a thousand thousands<\/td>\n<td>1,000,000<\/td>\n<td>\u00a0[latex]10^6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>billion<\/td>\n<td>(1000 x 1000) x 1000 = a thousand millions<\/td>\n<td>1,000,000,000<\/td>\n<td>\u00a0\u00a0[latex]10^9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>trillion<\/td>\n<td>(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\n<td>\u00a01,000,000,000,000<\/td>\n<td>\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>1 billion can be written as 1,000,000,000 or represented as\u00a0[latex]10^9[\/latex]. How would 2 billion be\u00a0represented? Since 2 billion is 2 times 1\u00a0billion, then 2 billion can be written as [latex]2\\times10^9[\/latex].<\/p>\n<p>A\u00a0light year is the number of miles light travels in one year, about 5,880,000,000,000. \u00a0That&#8217;s a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number.\u00a0Using scientific notation, the distance is [latex]5.88\\times10^{12}[\/latex]\u00a0miles. The exponent of 12 tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a\u00a0hydrogen atom, which is about 0.00000005 mm, and in scientific notation is\u00a0[latex]5\\times10^{-8}[\/latex]\u00a0mm. In this case the [latex]-8[\/latex] tells us how many places to count to the right of the decimal.<\/p>\n<p>Outlined in the box below are some important conventions of scientific notation format.<\/p>\n<div class=\"textbox shaded\">\n<h3>Scientific Notation<\/h3>\n<p>A positive number is written in scientific notation if it is written as [latex]a\\times10^{n}[\/latex]\u00a0where the coefficient <i>a<\/i>\u00a0is [latex]1\\leq{a}<10[\/latex], and <i>n <\/i>is an integer.<\/p>\n<\/div>\n<p>Look at the numbers below. Which of the numbers is written in scientific notation?<\/p>\n<table>\n<tbody>\n<tr>\n<td><b>Number<\/b><\/td>\n<td><b>Scientific Notation?<\/b><\/td>\n<td><b>Explanation<\/b><\/td>\n<\/tr>\n<tr>\n<td>[latex]1.85\\times10^{-2}[\/latex]<\/td>\n<td>yes<\/td>\n<td>[latex]1\\leq1.85<10[\/latex]\n\n[latex]-2[\/latex] is an integer<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle 1.083\\times {{10}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>no<\/td>\n<td>[latex]\\displaystyle \\frac{1}{2}[\/latex] is not an integer<\/td>\n<\/tr>\n<tr>\n<td>[latex]0.82\\times10^{14}[\/latex]<\/td>\n<td>no<\/td>\n<td>0.82 is not [latex]\\geq1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]10\\times10^{3}[\/latex]<\/td>\n<td>no<\/td>\n<td>10 is not &lt; 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now let\u2019s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.<\/p>\n<table>\n<tbody>\n<tr>\n<td colspan=\"2\">\n<p style=\"text-align: center;\"><b>Large Numbers<\/b><\/p>\n<\/td>\n<td><b>\u00a0<\/b><\/td>\n<td colspan=\"2\">\n<p style=\"text-align: center;\"><b>Small Numbers<\/b><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><b>Decimal Notation<\/b><\/td>\n<td><b>Scientific Notation<\/b><\/td>\n<td><b>\u00a0<\/b><\/td>\n<td><b>Decimal Notation<\/b><\/td>\n<td><b>Scientific Notation<\/b><\/td>\n<\/tr>\n<tr>\n<td>500.0<\/td>\n<td>[latex]5\\times10^{2}[\/latex]<\/td>\n<td><\/td>\n<td>0.05<\/td>\n<td>[latex]5\\times10^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>80,000.0<\/td>\n<td>[latex]8\\times10^{4}[\/latex]<\/td>\n<td><\/td>\n<td>0.0008<\/td>\n<td>[latex]8\\times10^{-4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>43,000,000.0<\/td>\n<td>[latex]4.3\\times10^{7}[\/latex]<\/td>\n<td><\/td>\n<td>0.00000043<\/td>\n<td>[latex]4.3\\times10^{-7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>62,500,000,000.0<\/td>\n<td>[latex]6.25\\times10^{10}[\/latex]<\/td>\n<td><\/td>\n<td>0.000000000625<\/td>\n<td>[latex]6.25\\times10^{-10}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<h3>Convert from decimal\u00a0notation to scientific\u00a0notation<\/h3>\n<p>To write a <i>large<\/i> number in scientific notation, move the decimal point to the left to obtain a number between 1 and 10. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 10 so that the expression has the same value.<\/p>\n<p>Let\u2019s look at an example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}180,000.=18,000.0\\times10^{1}\\\\1,800.00\\times10^{2}\\\\180.000\\times10^{3}\\\\18.0000\\times10^{4}\\\\1.80000\\times10^{5}\\\\180,000=1.8\\times10^{5}\\end{array}[\/latex]<\/p>\n<p>Notice that the decimal point was moved 5 places to the left, and the exponent is 5.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following numbers in scientific notation.<\/p>\n<ol>\n<li>[latex]920,000,000[\/latex]<\/li>\n<li>[latex]10,200,000[\/latex]<\/li>\n<li>[latex]100,000,000,000[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q628\">Show Solution<\/span><\/p>\n<div id=\"q628\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\underset{\\longleftarrow}{920,000,000}[\/latex] \u00a0We will move the decimal point to the left, it helps to place it\u00a0at the end of the number and then count how many times you move it to get one number before it that is between 1 and 10.\u00a0[latex]\\underset{\\longleftarrow}{920,000,000}=920,000,000.0[\/latex], move the decimal point 8 times to the left and you will have\u00a0[latex]9.20,000,000[\/latex], now we can replace the zeros with an exponent of 8,\u00a0[latex]9.2\\times10^{8}[\/latex]<\/li>\n<li>[latex]\\underset{\\longleftarrow}{10,200,000}=10,200,000.0=1.02\\times10^{7}[\/latex], note here how we included the 0 and the 2 after the decimal point. \u00a0In some disciplines, you may learn about when to include both of these. \u00a0Follow instructions from your teacher on rounding rules.<\/li>\n<li>[latex]\\underset{\\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\\times10^{11}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To write a small number (between 0 and 1) in scientific notation, you move the decimal to the <i>right<\/i> and the exponent will have to be <i>negative,\u00a0<\/i>as in the following example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underset{\\longrightarrow}{0.00004}=00.0004\\times10^{-1}\\\\000.004\\times10^{-2}\\\\0000.04\\times10^{-3}\\\\00000.4\\times10^{-4}\\\\000004.\\times10^{-5}\\\\0.00004=4\\times10^{-5}\\end{array}[\/latex]<\/p>\n<p>You may notice that the decimal point was moved five places to the <i>right <\/i>until you got \u00a0to the number 4, which is between 1 and 10. The exponent is [latex]\u22125[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following numbers in scientific notation.<\/p>\n<ol>\n<li>[latex]0.0000000000035[\/latex]<\/li>\n<li>[latex]0.0000000102[\/latex]<\/li>\n<li>[latex]0.00000000000000793[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q229054\">Show Solution<\/span><\/p>\n<div id=\"q229054\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\underset{\\longrightarrow}{0.0000000000035}=3.5\\times10^{-12}[\/latex], we moved the decimal 12 times to get to a number between 1 and 10<\/li>\n<li>[latex]\\underset{\\longrightarrow}{0.0000000102}=1.02\\times10^{-8}[\/latex]<\/li>\n<li>[latex]\\underset{\\longrightarrow}{0.00000000000000793}=7.93\\times10^{-15}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Examples:  Write a Number in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fsNu3AdIgdk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Convert from scientific notation to decimal notation<\/h3>\n<p>You can also write scientific notation as decimal notation. Recall\u00a0the number of miles that light travels in a year is [latex]5.88\\times10^{12}[\/latex], and a hydrogen atom has a diameter of [latex]5\\times10^{-8}[\/latex]\u00a0mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is<i> positive<\/i>, move the decimal point to the <i>right.<\/i> If the exponent is<i> negative<\/i>, move the decimal point to the <i>left.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5.88\\times10^{12}=\\underset{\\longrightarrow}{5.880000000000.}=5,880,000,000,000\\\\5\\times10^{-8}=\\underset{\\longleftarrow}{0.00000005.}=0.00000005\\end{array}[\/latex]<\/p>\n<p>For each power of 10, you move the decimal point one place. Be careful here and don\u2019t get carried away with the zeros\u2014the number of zeros after the decimal point will always be 1 <i>less<\/i> than the exponent because it takes one power of 10 to shift that first number to the left of the decimal.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following in decimal notation.<\/p>\n<ol>\n<li>[latex]4.8\\times10{-4}[\/latex]<\/li>\n<li>[latex]3.08\\times10^{6}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q489774\">Show Solution<\/span><\/p>\n<div id=\"q489774\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]4.8\\times10^{-4}[\/latex], the exponent is negative, so we need to move the decimal to the left. \u00a0[latex]\\underset{\\longleftarrow}{4.8\\times10^{-4}}=\\underset{\\longleftarrow}{.00048}[\/latex]<\/li>\n<li>[latex]3.08\\times10^{6}[\/latex], the exponent is positive, so we need to move the decimal to the right. \u00a0[latex]\\underset{\\longrightarrow}{3.08\\times10^{6}}=\\underset{\\longrightarrow}{3080000}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>To help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to wirte your ideas before you reveal the solution.<\/p>\n<p>1. You are writing a number that is greater than 1 in scientific notation. \u00a0Will your exponent be positive or negative?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>2. You are writing a number that is between 0 and 1 in scientific notation. \u00a0Will your exponent be positive or negative?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>3. What power do you need to put on 10 to get a result of 1?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q824936\">Show Solution<\/span><\/p>\n<div id=\"q824936\" class=\"hidden-answer\" style=\"display: none\">\n1. <em>You are writing a number that is greater than 1 in scientific notation. Will your exponent be positive or negative?<\/em> For numbers greater than 1, the exponent on 10 will be positive when you are using scientific notation. Refer to the table presented above:<\/p>\n<table class=\"undefined\">\n<thead>\n<tr>\n<td>Word<\/td>\n<td>How many thousands<\/td>\n<td>Number<\/td>\n<td>Scientific Notation<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>million<\/td>\n<td>1000 x 1000 = a thousand thousands<\/td>\n<td>1,000,000<\/td>\n<td>\u00a0[latex]10^6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>billion<\/td>\n<td>(1000 x 1000) x 1000 = a thousand millions<\/td>\n<td>1,000,000,000<\/td>\n<td>\u00a0\u00a0[latex]10^9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>trillion<\/td>\n<td>(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\n<td>\u00a01,000,000,000,000<\/td>\n<td>\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. <em>You are writing a number that is between 0 and 1 in scientific notation. Will your exponent be positive or negative?<\/em> We can reason that since numbers greater than 1 will have a positive exponent, numbers between\u00a00 and 1 will have a negative exponent. Why are we specifying numbers between 0 and 1? The numbers between 0 and 1 represent amounts that are fractional. Recall that we defined numbers with a negative exponent as\u00a0[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex], so if we have [latex]10^{-2}[\/latex] we have [latex]\\frac{1}{10\\times10}=\\frac{1}{100}[\/latex] which is a number between 0 and 1.<\/p>\n<p>3.\u00a0<em>What power do you need to put on 10 to get a result of 1?<\/em> Recall\u00a0that any number or variable with an exponent of 0 is equal to 1, as in this example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1\\\\\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}\\\\\\text{ therefore }\\\\{t}^{0}=1\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">We now have described the notation necessary to write all possible numbers on the number line in scientific notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the next video you will see how to convert a number written in scientific notation into decimal notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Examples:  Writing a Number in Decimal Notation When Given in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8BX0oKUMIjw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Multiplying and Dividing 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 10. 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 10 using the Product Rule. Add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]20.4\\times10^{-5}[\/latex]<\/p>\n<p>Convert 20.4 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 10 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 10 using the Product Rule\u2014add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]23.37\\times10^{-4}[\/latex]<\/p>\n<p>Convert 23.37 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 10 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<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-3\" 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<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 10 (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 10 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 0.82 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 10 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 10 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 10 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<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-4\" 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<h2 id=\"title3\">Solve application problems<\/h2>\n<div id=\"attachment_4384\" style=\"width: 371px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4384\" class=\"wp-image-4384\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26225033\/Screen-Shot-2016-05-26-at-3.50.25-PM-300x197.png\" alt=\"Molecule of water with one oxygen bonded to two hydrogen.\" width=\"361\" height=\"237\" \/><\/p>\n<p id=\"caption-attachment-4384\" class=\"wp-caption-text\">Water Molecule<\/p>\n<\/div>\n<p>Learning rules for exponents\u00a0seems pointless without context, so let&#8217;s explore some examples of using scientific notation that involve real problems. First, let&#8217;s look at an example of how scientific notation can be used to describe real measurements.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Match each length in the table with the appropriate number of meters described in scientific notation below.<\/p>\n<table>\n<tbody>\n<tr>\n<td>The height of a desk<\/td>\n<td>Diameter of\u00a0water molecule<\/td>\n<td>Diameter of Sun at its equator<\/td>\n<\/tr>\n<tr>\n<td>Distance from Earth to Neptune<\/td>\n<td>Diameter of Earth at the Equator<\/td>\n<td>Height of Mt. Everest (rounded)<\/td>\n<\/tr>\n<tr>\n<td>Diameter\u00a0of average human cell<\/td>\n<td>Diameter of a large grain of sand<\/td>\n<td>Distance a bullet travels in one second<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<thead>\n<tr>\n<td>Power of 10, units in meters<\/td>\n<td>Length from table above<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]10^{12}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{9}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{6}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{4}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{2}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{0}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-3}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-5}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-10}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993302\">Show Solution<\/span><\/p>\n<div id=\"q993302\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<thead>\n<tr>\n<td>Power of 10, units in meters<\/td>\n<td>Length from table above<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]10^{12}[\/latex]<\/td>\n<td>Distance from Earth to Neptune<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{9}[\/latex]<\/td>\n<td>Diameter of Sun at it&#8217;s Equator<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{6}[\/latex]<\/td>\n<td>Diameter of Earth at the Equator<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{4}[\/latex]<\/td>\n<td>Height of Mt. Everest (rounded)<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{2}[\/latex]<\/td>\n<td>Distance a bullet travels in one second<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{0}[\/latex]<\/td>\n<td>The height of a desk<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-3}[\/latex]<\/td>\n<td>Diameter of a large grain of sand<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-5}[\/latex]<\/td>\n<td>Diameter\u00a0of average human cell<\/td>\n<\/tr>\n<tr>\n<td>[latex]10^{-10}[\/latex]<\/td>\n<td>Diameter of\u00a0water molecule<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"attachment_4382\" style=\"width: 191px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4382\" class=\"wp-image-4382\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26224141\/Screen-Shot-2016-05-26-at-3.41.43-PM-300x210.png\" alt=\"Red Blood Cells.\" width=\"181\" height=\"126\" \/><\/p>\n<p id=\"caption-attachment-4382\" class=\"wp-caption-text\">Red Blood Cells<\/p>\n<\/div>\n<p>One of the most important parts of solving a &#8220;real&#8221; problem is translating the words into appropriate mathematical terms, and recognizing when a well known formula may help.\u00a0Here&#8217;s an example that requires you to find the density\u00a0of a cell, given its mass and volume. Cells aren&#8217;t visible to the naked eye, so their measurements, as described with scientific notation, involve negative exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Human cells come in a wide variety of shapes and sizes. The mass of an average human cell is about [latex]2\\times10^{-11}[\/latex] grams<a class=\"footnote\" title=\"Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https:\/\/en.wikipedia.org\/wiki\/Orders_of_magnitude_(mass)\" id=\"return-footnote-4333-1\" href=\"#footnote-4333-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>Red blood cells are one of the smallest types of cells<a class=\"footnote\" title=\"How Big is a Human Cell?\" id=\"return-footnote-4333-2\" href=\"#footnote-4333-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>, clocking in at a volume of approximately [latex]10^{-6}\\text{ meters }^3[\/latex].<a class=\"footnote\" title=\"How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from http:\/\/www.weizmann.ac.il\/plants\/Milo\/images\/humanCellSize120116Clean.pdf\" id=\"return-footnote-4333-3\" href=\"#footnote-4333-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a> Biologists have recently discovered how to use the density of some types of cells to indicate the presence of disorders such as sickle cell anemia or leukemia. <a class=\"footnote\" title=\"Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., &amp; Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073\/pnas.1104651108\" id=\"return-footnote-4333-4\" href=\"#footnote-4333-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a>\u00a0Density is calculated as the ratio of [latex]\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]. Calculate the density of an average human cell.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q856454\">Show Solution<\/span><\/p>\n<div id=\"q856454\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:\u00a0<\/strong>We are given an average cellular mass and volume as well as the formula for density. We are looking for the density of an average human cell.<\/p>\n<p><strong>Define and Translate:\u00a0<\/strong>\u00a0[latex]m=\\text{mass}=2\\times10^{-11}[\/latex], [latex]v=\\text{volume}=10^{-6}\\text{ meters}^3[\/latex], [latex]\\text{density}=\\frac{\\text{ mass }}{\\text{ volume }}[\/latex]<\/p>\n<p><strong>Write and Solve:\u00a0<\/strong>Use the quotient rule to simplify the ratio.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{ density }=\\frac{2\\times10^{-11}\\text{ grams }}{10^{-6}\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-11-\\left(-6\\right)}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\text{ }\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}\\\\\\end{array}[\/latex]<\/p>\n<p>If scientists know the density of healthy cells, they can compare the density of a sick person&#8217;s\u00a0cells to that to rule out or test for disorders or diseases that may affect cellular density.<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p style=\"text-align: left;\">The average density of a human cell is [latex]2\\times10^{-5}\\frac{\\text{ grams }}{\\text{ meters }^3}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<p>The following video provides an example of how to find the number of operations a computer can perform in a very short amount of time.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Application of Scientific Notation - Quotient 2 (Time for Computer Operations)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Cbm6ejEbu-o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"attachment_4381\" style=\"width: 352px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4381\" class=\"wp-image-4381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/26223918\/Screen-Shot-2016-05-26-at-3.39.02-PM-300x137.png\" alt=\"Earth in the foreground, sun in the background, light beams traveling from the sun to the earth.\" width=\"342\" height=\"156\" \/><\/p>\n<p id=\"caption-attachment-4381\" class=\"wp-caption-text\">Light traveling from the sun to the earth.<\/p>\n<\/div>\n<p>In the next example, you will use another well known formula, [latex]d=r\\cdot{t}[\/latex], to find how long it takes light to travel from the sun to the earth. Unlike the previous example, the distance between the earth and the sun is massive, so the numbers you will work with have positive exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The speed of light is [latex]3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}[\/latex]. If the sun is [latex]1.5\\times10^{11}[\/latex] meters from earth, how many seconds does it take for sunlight to reach the earth? \u00a0Write your answer in scientific notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q532092\">Show Solution<\/span><\/p>\n<div id=\"q532092\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:\u00a0<\/strong>We are looking for how long\u2014an amount of time. We are given a rate which has units of meters per second and a distance in meters. This is a [latex]d=r\\cdot{t}[\/latex] problem.<\/p>\n<p><strong>Define and Translate:\u00a0<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=1.5\\times10^{11}\\\\r=3\\times10^{8}\\frac{\\text{ meters }}{\\text{ second }}\\\\t=\\text{ ? }\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><strong>Write and Solve:\u00a0<\/strong>Substitute the values we are given into the\u00a0[latex]d=r\\cdot{t}[\/latex] equation. We will work without units to make it easier. Often, scientists will work with units to make sure they have made correct calculations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}d=r\\cdot{t}\\\\1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Divide both sides of the equation by\u00a0[latex]3\\times10^{8}[\/latex] to isolate\u00a0<em>t.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\\\\\text{ }\\\\\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with\u00a0<em>t.\u00a0<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{1.5\\times10^{11}}{3\\times10^{8}}=\\frac{3\\times10^{8}}{3\\times10^{8}}\\cdot{t}\\\\\\text{ }\\\\\\left(\\frac{1.5}{3}\\right)\\times\\left(\\frac{10^{11}}{10^{8}}\\right)=t\\\\\\text{ }\\\\\\left(0.5\\right)\\times\\left(10^{11-8}\\right)=t\\\\0.5\\times10^3=t\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">This answer is not in scientific notation, so we will move the decimal to the right, which means we need to subtract one factor of 10.<\/p>\n<p style=\"text-align: center;\">[latex]0.5\\times10^3=5.0\\times10^2=t[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The time it takes light to travel from the sun to the earth is [latex]5.0\\times10^2=t[\/latex] seconds, or in standard\u00a0notation, 500 seconds. \u00a0That&#8217;s not bad considering how far it has to travel!<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video\u00a0we calculate how many miles the participants of the New York marathon ran combined, and compare that to the circumference of the earth.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-6\" title=\"Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/15tw4-v100Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h3>Summary<\/h3>\n<p>Large and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.<\/p>\n<p>Scientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of 10. The format is written [latex]a\\times10^{n}[\/latex], where [latex]1\\leq{a}<10[\/latex]\u00a0and <i>n <\/i>is an integer.\u00a0To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.<\/p>\n<p>&nbsp;<\/p>\n<h2 id=\"video1\" class=\"no-indent\" style=\"text-align: left;\"><\/h2>\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-4333\">\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><li>Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/15tw4-v100Y\">https:\/\/youtu.be\/15tw4-v100Y<\/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>Application of Scientific Notation - Quotient 2 (Time for Computer Operations). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Cbm6ejEbu-o\">https:\/\/youtu.be\/Cbm6ejEbu-o<\/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>Screenshot: water molecule. <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><li>Screenshot: red blood cells. <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><li>Screenshot: light traveling from the sun to the earth. <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: Write a Number 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\/fsNu3AdIgdk\">https:\/\/youtu.be\/fsNu3AdIgdk<\/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: Writing a Number in Decimal Notation When Given 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\/8BX0oKUMIjw\">https:\/\/youtu.be\/8BX0oKUMIjw<\/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) . <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><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-4333-1\">Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https:\/\/en.wikipedia.org\/wiki\/Orders_of_magnitude_(mass) <a href=\"#return-footnote-4333-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-4333-2\">How Big is a Human Cell? <a href=\"#return-footnote-4333-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-4333-3\">How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from http:\/\/www.weizmann.ac.il\/plants\/Milo\/images\/humanCellSize120116Clean.pdf <a href=\"#return-footnote-4333-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-4333-4\">Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., &amp; Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073\/pnas.1104651108 <a href=\"#return-footnote-4333-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":21,"menu_order":4,"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 Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Examples: Write a Number in Scientific Notation\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/fsNu3AdIgdk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Examples: Writing a Number in Decimal Notation When Given in Scientific Notation\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/8BX0oKUMIjw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Examples: Dividing Numbers Written in Scientific Notation\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/RlZck2W5pO4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Examples: Multiplying Numbers Written in Scientific Notation\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5ZAY4OCkp7U\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Application of Scientific Notation - 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