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6.2: Scientific Notation

  • Page ID
    45059
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    One application of exponent properties is scientific notation. Scientific notation is used to represent really large or really small numbers, like the numbers that are too large or small to display on the calculator. For example, the distance light travels per year in miles is a very large number (\(5,879,000,000,000\)) and the mass of a single hydrogen atom in grams is a very small number (\(0.00000000000000000000000167\)). Basic operations, such as multiplication and division, with these numbers, would be quite cumbersome. However, the exponent properties allow us for simpler calculations.

    Definition: Scientific Notation

    Scientific notation is a notation for representing extremely large or small numbers in form of \[a\times 10^N,\nonumber\] where \(N\) is an integer, \(1 ≤ a < 10\), and \(N\) is number of decimal places from the right or left we moved to obtain \(a\).

    A few notes regarding scientific notation:

    • \(N\) is the way we convert between scientific and standard notation.
    • \(N\) represents the number of times we multiply or divide by \(10\). (Recall, multiplying by \(10\) moves the decimal point of a number one place value.)
    • We decide which direction to move the decimal (left or right) by remembering that in standard notation, positive exponents are numbers greater than ten and negative exponents are numbers less than one (but larger than zero).

    Case 1. If we move the decimal to the left with a number in standard notation, then \(N\) will be positive.

    Case 2. If we move the decimal to the right with a number in standard notation, then \(N\) will be negative.

    Convert Numbers to Scientific Notation

    Example 6.2.1

    Convert \(14,200\) to scientific notation.

    Solution

    Since this number is greater than \(10\), then we move the decimal to the left and \(N\) is positive. First we will find \(a\), then \(N\).

    \[\begin{array}{rl}14200\color{blue}{.0}\color{black}{}&\text{Identify the location of the decimal} \\1\color{blue}{.}\:\stackrel{\color{blue}{\curvearrowleft .}}{\color{black}{4\:}}\:\stackrel{\color{blue}{\curvearrowleft .}}{\color{black}{2}}\:\stackrel{\color{blue}{\curvearrowleft .}}{\color{black}{0}}\:\stackrel{\color{blue}{\curvearrowleft }}{\color{black}{0. 0}}&\text{Four decimal places to the left} \\ 1.42&\text{The value of }a\end{array}\nonumber\]

    Since we moved \(4\) decimal places to the left to obtain \(1.42\), then we know \(N = 4\), i.e., the exponent on the \(10\) is \(4\). Hence, rewriting \(14,200\) from standard notation to scientific notation, we get \[1.42\times 10^4\nonumber\]

    Note

    Be sure to always move the decimal however many decimal places to obtain a number between \(1\) and \(10\). In Example 6.2.1 , we only moved four decimal places because that is the number of decimal places we needed to move to obtain a number between \(1\) and \(10\).

    Example 6.2.2

    Convert \(0.0042\) to scientific notation.

    Solution

    Since this number is less than \(1\) (but greater than zero), then we move the decimal to the right and \(N\) is negative. First we will find \(a\), then \(N\).

    \[\begin{array}{rl}\color{blue}{0.}\color{black}{}0042&\text{Identify the location of the decimal} \\ \stackrel{\color{blue}{\curvearrowright\: .}}{\color{black}{0.\: 0}}\stackrel{\color{blue}{\:\curvearrowright\: .}}{\: \color{black}{0}} \stackrel{\color{blue}{\curvearrowright\: }}{\color{black}{\:4}} .2&\text{Three decimal places to the right} \\ 4.2&\text{The value for }a\end{array}\nonumber\]

    Since we moved \(3\) decimal places to the right to obtain \(4.2\), then we know \(N = −3\), i.e., the exponent on the \(10\) is \(−3\). Hence, rewriting \(0.0042\) from standard notation to scientific notation, we get \[4.2\times 10^{-3}\nonumber\]

    Convert Numbers from Scientific Notation to Standard Notation

    Convert Numbers from Scientific Notation to Standard Notation

    To convert a number from scientific notation of the form \[a\times 10^{N}\nonumber\] to standard notation, we can follow these rules of thumb.

    • If \(N\) is positive, this means the original number was greater than \(10\), we move the decimal to the right \(N\) times.
    • If \(N\) is negative, this means the original number was less than \(1\) (but greater than zero), we move the decimal to the left \(N\) times.
    Example 6.2.3

    Convert \(3.21\times 10^5\) to standard notation.

    Solution

    Since \(N = 5\), which is positive, then this means the standard notation of the number is greater than \(10\) and we move the decimal to the right \(5\) times.

    \[\begin{array}{rl}\color{blue}{3.}\color{black}{}21&\text{Identify the location of the decimal} \\ \stackrel{\color{blue}{\curvearrowright .}}{\color{black}{3.\: 2}}\stackrel{\color{blue}{\curvearrowright .}}{\: \color{black}{1}}\stackrel{\color{blue}{\curvearrowright .}}{\:\color{blue}{0}}\stackrel{\color{blue}{\curvearrowright .}}{\:\color{blue}{0}}\stackrel{\color{blue}{\curvearrowright }}{\:\color{blue}{0}}&\text{Five decimal places to the right} \\ 321000.&\text{Standard notation}\end{array}\nonumber\]

    Since we moved \(5\) decimal places to the right to obtain \(321,000\), notice, as we were moving the decimal, there were place values with no digits and so we wrote in the zeros. In general, we do this when there are place values with no digits when expanding the numbers.

    Example 6.2.4

    Convert \(7.4\times 10^{-3}\) to standard notation.

    Solution

    Since \(N = −3\), which is negative, then this means the standard notation of the number is less than \(1\) (but greater than zero) and we move the decimal to the left \(3\) times.

    \[\begin{array}{rl}\color{blue}{7.}\color{black}{4}&\text{Identify the location of the decimal} \\ \stackrel{\color{blue}{\curvearrowleft .}}{\color{blue}{0.0}}\stackrel{\color{blue}{\curvearrowleft .}}{\color{blue}{\: 0}}\stackrel{\color{blue}{\curvearrowleft}}{\color{black}{7.}}4&\text{Three decimal places to the left} \\ 0.0074&\text{Standard notation}\end{array}\nonumber\]

    Since we moved \(3\) decimal places to the left to obtain \(0.0074\), notice we had to write zeros in for the tenths and hundredths place.

    Multiply and Divide Numbers in Scientific Notation

    Converting numbers between standard notation and scientific notation is important in understanding scientific notation and its purpose. Next, we multiply and divide numbers in scientific notation using the exponent properties. If the immediate result isn’t written in scientific notation, we will complete an additional step in writing the answer in scientific notation.

    Steps for multiplying and dividing numbers in scientific notation

    Step 1. Rewrite the factors as multiplying or dividing \(a\)-values and then multiplying or dividing \(10^N\) values.

    Step 2. Multiply or divide the \(a\) values and apply the product or quotient rule of exponents to add or subtract the exponents, \(N\), on the base \(10\)s, respectively.

    Step 3. Be sure the result is in scientific notation. If not, then rewrite in scientific notation.

    Example 6.2.5

    Multiply: \((2.1\times 10^{-7})(3.7\times 10^5)\)

    Solution

    Step 1. Rewrite the factors as multiplying \(a\)-values and then multiplying \(10^N\) values. \[(2.1)(3.7)\times (10^{-7}\cdot 10^5)\nonumber\]

    Step 2. Multply \(a\) values and apply the product rule of exponents on the \(10^N\) values. \[\begin{array}{rl}(2.1)(3.7)\times 10^{-7+5}&\text{Simplify} \\ 7.77\times 10^{-2}&\text{Product}\end{array}\nonumber\]

    Step 3. Since the product resulted in scientific notation, we leave it as is.

    Example 6.2.6

    Divide: \(\dfrac{4.96\times 10^4}{3.1\times 10^{-3}}\)

    Solution

    Step 1. Rewrite the factors as dividing \(a\)-values and then dividing \(10^N\) values. \[\dfrac{4.96}{3.1}\times \dfrac{10^4}{10^{-3}}\nonumber\]

    Step 2. Multply \(a\) values and apply the quotient rule of exponents on the \(10^N\) values. \[\begin{array}{rl}\dfrac{4.96}{3.1}\times 10^{4-(-3)}&\text{Simplify} \\ 1.6\times 10^7&\text{Quotient}\end{array}\nonumber\]

    Step 3. Since the quotient resulted in scientific notation, we leave it as is.

    Example 6.2.7

    Multiply: \((4.7\times 10^{-3})(6.1\times 10^9)\)

    Solution

    Step 1. Rewrite the factors as multiplying \(a\)-values and then multiplying \(10^N\) values. \[(4.7)(6.1)\times (10^{-3}\times 10^9)\nonumber\]

    Step 2. Multiply \(a\) values and apply the product rule of exponents on the \(10^N\) values. \[\begin{array}{rl}(4.7)(6.1)\times 10^{-3+9}&\text{Simplify} \\ 28.67\times 10^6&\text{Product}\end{array}\nonumber\]

    Step 3. Since the product resulted a number not in scientific notation, we have rewrite it so that it is in scientific notation. Hence, we need \(a\) to be a number at least \(1\) and less than \(10\), and \(28.67\) is greater than \(10\), then we move the decimal to the left and \(N\) is positive. \[\begin{array}{rl}(2\color{blue}{8.}\color{black}{}67)\times 10^6&\text{Identify the location of the decimal} \\ \left( \stackrel{\color{blue}{\curvearrowleft}}{\color{black}{2}\color{blue}{.8}}\color{black}{}.67\times 10^1\right)\times 10^6&\text{One decimal place to the left} \\ 2.867\times 10^1\times 10^6&\text{Apply product rule of exponents} \\ 2.867\times 10^7&\text{Scientific notation}\end{array}\nonumber\]

    Note

    Archimedes (287 BC-212 BC), the Greek mathematician, developed a system for representing large numbers using a system very similar to scientific notation. He used his system to calculate the number of grains of sand it would take to fill the universe. His conclusion was \(10^{63}\) grains of sand because he figured the universe to have a diameter of \(10^{14}\) stadia or about \(2\) light years.

    Example 6.2.8

    Divide: \(\dfrac{2.014\times 10^{-3}}{3.8\times 10^{-7}}\)

    Solution

    Step 1. Rewrite the factors as dividing \(a\)-values and then dividing \(10^N\) values. \[\dfrac{2.014}{3.8}\times \dfrac{10^{-3}}{10^{-7}}\nonumber\]

    Step 2. Divide \(a\) values and apply the quotient rule of exponents on the \(10^N\) values. \[\begin{array}{rl}\dfrac{2.014}{3.8}\times 10^{-3-(-7)}&\text{Simplify} \\ 0.53\times 10^4 &\text{Quotient}\end{array}\nonumber\]

    Step 3. Since the quotient resulted a number not in scientific notation, we have rewrite it so that it is in scientific notation. Hence, we need \(a\) to be a number at least \(1\) and less than \(10\), and \(0.53\) is less than \(1\) (but greater than zero), then we move the decimal to the right and \(N\) is negative. \[\begin{array}{rl}(\color{blue}{0.}\color{black}{}53)\times 10^4 &\text{Identify the location of the decimal} \\ \left(0.\color{blue}{\stackrel{\curvearrowright}{5}}\color{black}{}.3\times 10^{-1}\right)\times 10^4&\text{One decimal place to the right} \\ 5.3\times 10^{-1}\times 10^4&\text{Apply product rule of exponents} \\ 5.3\times 10^3&\text{Scientific notation}\end{array}\nonumber\]

    Scientific Notation Homework

    Write each number in scientific notation

    Exercise 6.2.1

    \(885\)

    Exercise 6.2.2

    \(0.081\)

    Exercise 6.2.3

    \(0.039\)

    Exercise 6.2.4

    \(0.000744\)

    Exercise 6.2.5

    \(1.09\)

    Exercise 6.2.6

    \(15,000\)

    Write each number in standard notation.

    Exercise 6.2.7

    \(8.7\times 10^5\)

    Exercise 6.2.8

    \(9\times 10^{-4}\)

    Exercise 6.2.9

    \(2\times 10^0\)

    Exercise 6.2.10

    \(2.56\times 10^2\)

    Exercise 6.2.11

    \(5\times 10^4\)

    Exercise 6.2.12

    \(6\times 10^{-5}\)

    Simplify. Write each answer in scientific notation.

    Exercise 6.2.13

    \((7\times 10^{-1})(2\times 10^{-3})\)

    Exercise 6.2.14

    \((5.26\times 10^{-5})(3.16\times 10^{-2})\)

    Exercise 6.2.15

    \((2.6\times 10^{-2})(6\times 10^{-2})\)

    Exercise 6.2.16

    \(\dfrac{4.9\times 10^1}{2.7\times 10^{-3}}\)

    Exercise 6.2.17

    \(\dfrac{5.33\times 10^{-6}}{9.62\times 10^{-2}}\)

    Exercise 6.2.18

    \((5.5\times 10^{-5})^2\)

    Exercise 6.2.19

    \((7.8\times 10^{-2})^5\)

    Exercise 6.2.20

    \((8.03\times 10^4)^{-4}\)

    Exercise 6.2.21

    \(\dfrac{6.1\times 10^{-6}}{5.1\times 10^{-4}}\)

    Exercise 6.2.22

    \((3.6\times 10^0)(6.1\times 10^{-3})\)

    Exercise 6.2.23

    \((1.8\times 10^{-5})^{-3}\)

    Exercise 6.2.24

    \(\dfrac{9\times 10^4}{7.83\times 10^{-2}}\)

    Exercise 6.2.25

    \(\dfrac{3.22\times 10^{-3}}{7\times 10^{-6}}\)

    Exercise 6.2.26

    \(\dfrac{2.4\times 10^{-6}}{6.5\times 10^0}\)

    Exercise 6.2.27

    \(\dfrac{6\times 10^3}{5.8\times 10^{-3}}\)

    Exercise 6.2.28

    \((2\times 10^{-6})(8.8\times 10^{-5})\)

    Exercise 6.2.29

    \((5.1\times 10^6)(9.84\times 10^{-1})\)

    Exercise 6.2.30

    \(\dfrac{7.4\times 10^4}{1.7\times 10^{-4}}\)

    Exercise 6.2.31

    \(\dfrac{7.2\times 10^{-1}}{7.32\times 10^{-1}}\)

    Exercise 6.2.32

    \(\dfrac{3.2\times 10^{-3}}{5.02\times 10^0}\)

    Exercise 6.2.33

    \((9.6\times 10^3)^{-4}\)

    Exercise 6.2.34

    \((5.4\times 10^6)^{-3}\)

    Exercise 6.2.35

    \((6.88\times 10^{-4})(4.23\times 10^1)\)

    Exercise 6.2.36

    \(\dfrac{8.4\times 10^5}{7\times 10^{-2}}\)

    Exercise 6.2.37

    \((3.15\times 10^3)(8\times 10^{-1})\)

    Exercise 6.2.38

    \(\dfrac{9.58\times 10^{-2}}{1.14\times 10^{-3}}\)

    Exercise 6.2.39

    \((8.3\times 10^1)^5\)

    Exercise 6.2.40

    \(\dfrac{5\times 10^6}{6.69\times 10^2}\)

    Exercise 6.2.41

    \((9\times 10^{-2})^{-3}\)

    Exercise 6.2.42

    \((2\times 10^4)(6\times 10^1)\)


    This page titled 6.2: Scientific Notation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.