{"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&p=4333"},"modified":"2021-06-02T22:50:41","modified_gmt":"2021-06-02T22:50:41","slug":"outcome-writing-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/outcome-writing-scientific-notation\/","title":{"raw":"5.3: Scientific Notation","rendered":"5.3: Scientific Notation"},"content":{"raw":"
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Section 5.3 Learning Objectives<\/h3>\r\n5.3: Scientific Notation<\/strong>\r\n
    \r\n \t
  • Convert from decimal notation to scientific notation<\/li>\r\n \t
  • Convert from scientific notation to decimal notation<\/li>\r\n \t
  • Find the product of numbers written in scientific notation and write in appropriate scientific notation form<\/li>\r\n \t
  • Find the quotient of numbers written in scientific notation and write in appropriate scientific notation form<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    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.\u00a0S<\/strong>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 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\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Word<\/td>\r\nHow many thousands<\/td>\r\nNumber<\/td>\r\nScientific Notation<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
    million<\/td>\r\n1000 x 1000 = a thousand thousands<\/td>\r\n1,000,000<\/td>\r\n\u00a0[latex]10^6[\/latex]<\/td>\r\n<\/tr>\r\n
    billion<\/td>\r\n(1000 x 1000) x 1000 = a thousand millions<\/td>\r\n1,000,000,000<\/td>\r\n\u00a0\u00a0[latex]10^9[\/latex]<\/td>\r\n<\/tr>\r\n
    trillion<\/td>\r\n(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\r\n\u00a01,000,000,000,000<\/td>\r\n\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 is how many places to count to the left of the decimal in 5,880,000,000,000 to convert it into scientific notation. 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] is how many places to count to the right of the decimal in 0.00000005 to convert it into scientific notation.\r\n\r\nOutlined in the box below are some important conventions of scientific notation format.\r\n
    \r\n

    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 a<\/i>\u00a0is [latex]1\\leq{a}<10[\/latex], and 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\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Number<\/b><\/td>\r\nScientific Notation?<\/b><\/td>\r\nExplanation<\/b><\/td>\r\n<\/tr>\r\n
    [latex]1.85\\times10^{-2}[\/latex]<\/td>\r\nyes<\/td>\r\n[latex]1\\leq1.85<10[\/latex]\r\n\r\n[latex]-2[\/latex] is an integer<\/td>\r\n<\/tr>\r\n
    [latex] \\displaystyle 1.083\\times {{10}^{\\frac{1}{2}}}[\/latex]<\/td>\r\nno<\/td>\r\n[latex] \\displaystyle \\frac{1}{2}[\/latex] is not an integer<\/td>\r\n<\/tr>\r\n
    [latex]0.82\\times10^{14}[\/latex]<\/td>\r\nno<\/td>\r\n0.82 is not [latex]\\geq1[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]10\\times10^{3}[\/latex]<\/td>\r\nno<\/td>\r\n10 is not < 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\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    \r\n

    Large Numbers<\/b><\/p>\r\n<\/td>\r\n

    		\u00a0<\/b><\/td>\r\n\r\n

    Small Numbers<\/b><\/p>\r\n<\/td>\r\n<\/tr>\r\n

    Decimal Notation<\/b><\/td>\r\nScientific Notation<\/b><\/td>\r\n\u00a0<\/b><\/td>\r\nDecimal Notation<\/b><\/td>\r\nScientific Notation<\/b><\/td>\r\n<\/tr>\r\n
    500.0<\/td>\r\n[latex]5\\times10^{2}[\/latex]<\/td>\r\n<\/td>\r\n0.05<\/td>\r\n[latex]5\\times10^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n
    80,000.0<\/td>\r\n[latex]8\\times10^{4}[\/latex]<\/td>\r\n<\/td>\r\n0.0008<\/td>\r\n[latex]8\\times10^{-4}[\/latex]<\/td>\r\n<\/tr>\r\n
    43,000,000.0<\/td>\r\n[latex]4.3\\times10^{7}[\/latex]<\/td>\r\n<\/td>\r\n0.00000043<\/td>\r\n[latex]4.3\\times10^{-7}[\/latex]<\/td>\r\n<\/tr>\r\n
    62,500,000,000.0<\/td>\r\n[latex]6.25\\times10^{10}[\/latex]<\/td>\r\n<\/td>\r\n0.000000000625<\/td>\r\n[latex]6.25\\times10^{-10}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

    Convert from decimal\u00a0notation to scientific\u00a0notation<\/h3>\r\nTo write a 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

    [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

    \r\n

    Example 1<\/h3>\r\nWrite the following numbers in scientific notation.\r\n
      \r\n \t
    1. [latex]920,000,000[\/latex]<\/li>\r\n \t
    2. [latex]10,200,000[\/latex]<\/li>\r\n \t
    3. [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
        \r\n \t
      1. [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
      2. [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
      3. [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 \r\n\r\nTo write a small number (between 0 and 1) in scientific notation, you move the decimal to the right<\/i> and the exponent will have to be negative,\u00a0<\/i>as in the following example.\r\n

        [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 right <\/i>until you got \u00a0to the number 4, which is between 1 and 10. The exponent is [latex]\u22125[\/latex].\r\n

        \r\n

        Example 2<\/h3>\r\nWrite the following numbers in scientific notation.\r\n
          \r\n \t
        1. [latex]0.0000000000035[\/latex]<\/li>\r\n \t
        2. [latex]0.0000000102[\/latex]<\/li>\r\n \t
        3. [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
            \r\n \t
          1. [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
          2. [latex]\\underset{\\longrightarrow}{0.0000000102}=1.02\\times10^{-8}[\/latex]<\/li>\r\n \t
          3. [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

            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 positive<\/i>, move the decimal point to the right.<\/i> If the exponent is negative<\/i>, move the decimal point to the left.<\/i>\r\n

            [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!\r\n

            \r\n

            Example 3<\/h3>\r\nWrite the following in decimal notation.\r\n
              \r\n \t
            1. [latex]4.8\\times10^{-4}[\/latex]<\/li>\r\n \t
            2. [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
                \r\n \t
              1. [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
              2. [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
                \r\n

                Think About It 1<\/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. 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\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                Word<\/td>\r\nHow many thousands<\/td>\r\nNumber<\/td>\r\nScientific Notation<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
                million<\/td>\r\n1000 x 1000 = a thousand thousands<\/td>\r\n1,000,000<\/td>\r\n\u00a0[latex]10^6[\/latex]<\/td>\r\n<\/tr>\r\n
                billion<\/td>\r\n(1000 x 1000) x 1000 = a thousand millions<\/td>\r\n1,000,000,000<\/td>\r\n\u00a0\u00a0[latex]10^9[\/latex]<\/td>\r\n<\/tr>\r\n
                trillion<\/td>\r\n(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\r\n\u00a01,000,000,000,000<\/td>\r\n\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. 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.\u00a0What 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

                [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

                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 \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

                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 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
                \r\n

                Example 4<\/h3>\r\n

                [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

                [latex]\\left(3\\times6.8\\right)\\left(10^{8}\\times10^{-13}\\right)[\/latex]<\/p>\r\nMultiply the coefficients.\r\n

                [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

                [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

                [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

                [latex]2.04\\times\\left(10^{1}\\times10^{-5}\\right)[\/latex]<\/p>\r\nMultiply using the Product Rule\u2014add the exponents.\r\n

                [latex]2.04\\times10^{1+\\left(-5\\right)}[\/latex]<\/p>\r\n\r\n

                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\nIn the following video you will see an example of how to multiply two 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 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
                \r\n

                Example 5<\/h3>\r\n

                [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

                [latex] \\displaystyle \\left( \\frac{2.829}{3.45} \\right)\\left( \\frac{{{10}^{-9}}}{{{10}^{-3}}} \\right)[\/latex]<\/p>\r\nDivide the coefficients.\r\n

                [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

                [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

                [latex]\\left(8.2\\times10^{-1}\\right)\\times10^{-6}[\/latex]<\/p>\r\nGroup the powers of 10 together using the associative property.\r\n

                [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

                [latex]8.2\\times10^{-1+\\left(-6\\right)}[\/latex]<\/p>\r\n\r\n

                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
                \r\n

                Think About it 2<\/h3>\r\n

                [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

                [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

                [latex] \\displaystyle \\frac{13.4945\\times {{10}^{10}}}{5.0\\times {{10}^{12}}}[\/latex]<\/p>\r\nRegroup using the associative property.\r\n

                [latex] \\displaystyle \\left( \\frac{13.4945}{5.0} \\right)\\left( \\frac{{{10}^{10}}}{{{10}^{12}}} \\right)[\/latex]<\/p>\r\nDivide the numbers.\r\n

                [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

                [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

                Answer<\/h4>\r\n[latex] \\displaystyle \\frac{\\left( 1.37\\times {{10}^{4}} \\right)\\left( 9.85\\times {{10}^{6}} \\right)}{5.0\\times {{10}^{12}}}=2.6989\\times {{10}^{-2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\nNotice that when you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator. You will see another example of dividing numbers written in scientific notation in the following video.\r\n\r\nhttps:\/\/youtu.be\/RlZck2W5pO4\r\n

                Solve an application problem<\/h2>\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
                \r\n

                Example 6<\/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\nRead 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\nDefine and Translate:\u00a0<\/strong>\r\n

                [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

                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

                [latex]\\begin{array}{c}d=r\\cdot{t}\\\\1.5\\times10^{11}=3\\times10^{8}\\cdot{t}\\end{array}[\/latex]<\/p>\r\n

                Divide both sides of the equation by\u00a0[latex]3\\times10^{8}[\/latex] to isolate\u00a0t.<\/em><\/p>\r\n

                [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

                On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with\u00a0t.\u00a0<\/em><\/p>\r\n

                [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

                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

                [latex]0.5\\times10^3=5.0\\times10^2=t[\/latex]<\/p>\r\n\r\n

                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\n

                Summary<\/h3>\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}<10[\/latex]\u00a0and n <\/i>is an integer.\r\n\r\nPerforming mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents. To 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 \r\n

                <\/h2>","rendered":"
                \n

                Section 5.3 Learning Objectives<\/h3>\n

                5.3: Scientific Notation<\/strong><\/p>\n

                  \n
                • Convert from decimal notation to scientific notation<\/li>\n
                • Convert from scientific notation to decimal notation<\/li>\n
                • Find the product of numbers written in scientific notation and write in appropriate scientific notation form<\/li>\n
                • Find the quotient of numbers written in scientific notation and write in appropriate scientific notation form<\/li>\n<\/ul>\n<\/div>\n

                   <\/p>\n

                  Convert between scientific and decimal notation<\/h2>\n

                  Before we can convert between scientific and decimal notation, we need to know the difference between the two.\u00a0S<\/strong>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

                  When a number is written in scientific notation, the 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\n\n\n\n\n\n\n
                  Word<\/td>\nHow many thousands<\/td>\nNumber<\/td>\nScientific Notation<\/td>\n<\/tr>\n<\/thead>\n
                  million<\/td>\n1000 x 1000 = a thousand thousands<\/td>\n1,000,000<\/td>\n\u00a0[latex]10^6[\/latex]<\/td>\n<\/tr>\n
                  billion<\/td>\n(1000 x 1000) x 1000 = a thousand millions<\/td>\n1,000,000,000<\/td>\n\u00a0\u00a0[latex]10^9[\/latex]<\/td>\n<\/tr>\n
                  trillion<\/td>\n(1000 x 1000 x 1000) x 1000 = a thousand billions<\/td>\n\u00a01,000,000,000,000<\/td>\n\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  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

                  A\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 is how many places to count to the left of the decimal in 5,880,000,000,000 to convert it into scientific notation. 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] is how many places to count to the right of the decimal in 0.00000005 to convert it into scientific notation.<\/p>\n

                  Outlined in the box below are some important conventions of scientific notation format.<\/p>\n

                  \n

                  Scientific Notation<\/h3>\n

                  A positive number is written in scientific notation if it is written as [latex]a\\times10^{n}[\/latex]\u00a0where the coefficient a<\/i>\u00a0is [latex]1\\leq{a}<10[\/latex], and n <\/i>is an integer.<\/p>\n<\/div>\n

                  Look at the numbers below. Which of the numbers is written in scientific notation?<\/p>\n\n\n\n\n\n\n\n
                  Number<\/b><\/td>\nScientific Notation?<\/b><\/td>\nExplanation<\/b><\/td>\n<\/tr>\n
                  [latex]1.85\\times10^{-2}[\/latex]<\/td>\nyes<\/td>\n[latex]1\\leq1.85<10[\/latex]\n\n[latex]-2[\/latex] is an integer<\/td>\n<\/tr>\n
                  [latex]\\displaystyle 1.083\\times {{10}^{\\frac{1}{2}}}[\/latex]<\/td>\nno<\/td>\n[latex]\\displaystyle \\frac{1}{2}[\/latex] is not an integer<\/td>\n<\/tr>\n
                  [latex]0.82\\times10^{14}[\/latex]<\/td>\nno<\/td>\n0.82 is not [latex]\\geq1[\/latex]<\/td>\n<\/tr>\n
                  [latex]10\\times10^{3}[\/latex]<\/td>\nno<\/td>\n10 is not < 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  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\n\n\n\n\n\n\n\n
                  \n

                  Large Numbers<\/b><\/p>\n<\/td>\n

                  		\u00a0<\/b><\/td>\n\n

                  Small Numbers<\/b><\/p>\n<\/td>\n<\/tr>\n

                  Decimal Notation<\/b><\/td>\nScientific Notation<\/b><\/td>\n\u00a0<\/b><\/td>\nDecimal Notation<\/b><\/td>\nScientific Notation<\/b><\/td>\n<\/tr>\n
                  500.0<\/td>\n[latex]5\\times10^{2}[\/latex]<\/td>\n<\/td>\n0.05<\/td>\n[latex]5\\times10^{-2}[\/latex]<\/td>\n<\/tr>\n
                  80,000.0<\/td>\n[latex]8\\times10^{4}[\/latex]<\/td>\n<\/td>\n0.0008<\/td>\n[latex]8\\times10^{-4}[\/latex]<\/td>\n<\/tr>\n
                  43,000,000.0<\/td>\n[latex]4.3\\times10^{7}[\/latex]<\/td>\n<\/td>\n0.00000043<\/td>\n[latex]4.3\\times10^{-7}[\/latex]<\/td>\n<\/tr>\n
                  62,500,000,000.0<\/td>\n[latex]6.25\\times10^{10}[\/latex]<\/td>\n<\/td>\n0.000000000625<\/td>\n[latex]6.25\\times10^{-10}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                   <\/p>\n

                  Convert from decimal\u00a0notation to scientific\u00a0notation<\/h3>\n

                  To write a 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

                  Let\u2019s look at an example.<\/p>\n

                  [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

                  Notice that the decimal point was moved 5 places to the left, and the exponent is 5.<\/p>\n

                  \n

                  Example 1<\/h3>\n

                  Write the following numbers in scientific notation.<\/p>\n

                    \n
                  1. [latex]920,000,000[\/latex]<\/li>\n
                  2. [latex]10,200,000[\/latex]<\/li>\n
                  3. [latex]100,000,000,000[\/latex]<\/li>\n<\/ol>\n
                    Show Solution<\/span><\/p>\n
                    \n
                      \n
                    1. [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
                    2. [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
                    3. [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>\n

                      To write a small number (between 0 and 1) in scientific notation, you move the decimal to the right<\/i> and the exponent will have to be negative,\u00a0<\/i>as in the following example.<\/p>\n

                      [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

                      You may notice that the decimal point was moved five places to the right <\/i>until you got \u00a0to the number 4, which is between 1 and 10. The exponent is [latex]\u22125[\/latex].<\/p>\n

                      \n

                      Example 2<\/h3>\n

                      Write the following numbers in scientific notation.<\/p>\n

                        \n
                      1. [latex]0.0000000000035[\/latex]<\/li>\n
                      2. [latex]0.0000000102[\/latex]<\/li>\n
                      3. [latex]0.00000000000000793[\/latex]<\/li>\n<\/ol>\n
                        Show Solution<\/span><\/p>\n
                        \n
                          \n
                        1. [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
                        2. [latex]\\underset{\\longrightarrow}{0.0000000102}=1.02\\times10^{-8}[\/latex]<\/li>\n
                        3. [latex]\\underset{\\longrightarrow}{0.00000000000000793}=7.93\\times10^{-15}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

                          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