Multiplying and Dividing Numbers in Scientific Notation
Learning Outcomes
- Multiply numbers expressed in scientific notation
- Divide numbers expressed in scientific notation
Multiplying Numbers Expressed in Scientific Notation
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’t powers of 10 (the a in [latex]a\times10^{n}[/latex]). Then multiply the powers of ten by adding the exponents. This will produce a new number times a different power of [latex]10[/latex]. All you have to do is check to make sure this new value is in scientific notation. If it isn’t, you convert it. Let’s look at some examples.Example
[latex]\left(3\times10^{8}\right)\left(6.8\times10^{-13}\right)[/latex]
Answer: Regroup using the commutative and associative properties.
[latex]\left(3\times6.8\right)\left(10^{8}\times10^{-13}\right)[/latex]
Multiply the coefficients.[latex]\left(20.4\right)\left(10^{8}\times10^{-13}\right)[/latex]
Multiply the powers of [latex]10[/latex] using the Product Rule. Add the exponents.[latex]20.4\times10^{-5}[/latex]
Convert [latex]20.4[/latex] into scientific notation by moving the decimal point one place to the left and multiplying by [latex]10^{1}[/latex].[latex]\left(2.04\times10^{1}\right)\times10^{-5}[/latex]
Group the powers of [latex]10[/latex] using the associative property of multiplication.[latex]2.04\times\left(10^{1}\times10^{-5}\right)[/latex]
Multiply using the Product Rule—add the exponents.[latex]2.04\times10^{1+\left(-5\right)}[/latex]
Answer
[latex-display]\left(3\times10^{8}\right)\left(6.8\times10^{-13}\right)=2.04\times10^{-4}[/latex-display]Example
[latex]\left(8.2\times10^{6}\right)\left(1.5\times10^{-3}\right)\left(1.9\times10^{-7}\right)[/latex]
Answer: Regroup using the commutative and associative properties.
[latex]\left(8.2\times1.5\times1.9\right)\left(10^{6}\times10^{-3}\times10^{-7}\right)[/latex]
Multiply the numbers.[latex]\left(23.37\right)\left(10^{6}\times10^{-3}\times10^{-7}\right)[/latex]
Multiply the powers of [latex]10[/latex] using the Product Rule—add the exponents.[latex]23.37\times10^{-4}[/latex]
Convert [latex]23.37[/latex] into scientific notation by moving the decimal point one place to the left and multiplying by [latex]10^{1}[/latex].[latex]\left(2.337\times10^{1}\right)\times10^{-4}[/latex]
Group the powers of [latex]10[/latex] using the associative property of multiplication.[latex]2.337\times\left(10^{1}\times10^{-4}\right)[/latex]
Multiply using the Product Rule and add the exponents.[latex]2.337\times10^{1+\left(-4\right)}[/latex]
Answer
[latex-display]\left(8.2\times10^{6}\right)\left(1.5\times10^{-3}\right)\left(1.9\times10^{-7}\right)=2.337\times10^{-3}[/latex-display]example
Multiply. Write answers in decimal form: [latex]\left(4\times {10}^{5}\right)\left(2\times {10}^{-7}\right)[/latex].Answer: Solution
| [latex]\left(4\times {10}^{5}\right)\left(2\times {10}^{-7}\right)[/latex] | |
| Use the Commutative Property to rearrange the factors. | [latex]4\cdot 2\cdot {10}^{5}\cdot {10}^{-7}[/latex] |
| Multiply [latex]4[/latex] by [latex]2[/latex] and use the Product Property to multiply [latex]{10}^{5}[/latex] by [latex]{10}^{-7}[/latex]. | [latex]8\times {10}^{-2}[/latex] |
| Change to decimal form by moving the decimal two places left. | [latex]{\Large\frac{8}{100}} = 0.08[/latex] |
try it
[ohm_question]146318[/ohm_question] [ohm_question]2826[/ohm_question]Dividing Numbers Expressed in Scientific Notation
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’t powers of [latex]10[/latex] (the a in [latex]a\times10^{n}[/latex]. Then you divide the powers of ten by subtracting the exponents. This will produce a new number times a different power of 10. If it isn’t already in scientific notation, you convert it, and then you’re done. Let’s look at some examples.Example
[latex] \displaystyle \frac{2.829\times 1{{0}^{-9}}}{3.45\times 1{{0}^{-3}}}[/latex]
Answer: Regroup using the associative property.
[latex] \displaystyle \left( \frac{2.829}{3.45} \right)\left( \frac{{{10}^{-9}}}{{{10}^{-3}}} \right)[/latex]
Divide the coefficients.[latex] \displaystyle \left(0.82\right)\left( \frac{{{10}^{-9}}}{{{10}^{-3}}} \right)[/latex]
Divide the powers of [latex]10[/latex] using the Quotient Rule. Subtract the exponents.[latex]\begin{array}{l}0.82\times10^{-9-\left(-3\right)}\\0.82\times10^{-6}\end{array}[/latex]
Convert [latex]0.82[/latex] into scientific notation by moving the decimal point one place to the right and multiplying by [latex]10^{-1}[/latex].[latex]\left(8.2\times10^{-1}\right)\times10^{-6}[/latex]
Group the powers of [latex]10[/latex] together using the associative property.[latex]8.2\times\left(10^{-1}\times10^{-6}\right)[/latex]
Multiply the powers of [latex]10[/latex] using the Product Rule—add the exponents.[latex]8.2\times10^{-1+\left(-6\right)}[/latex]
Answer
[latex-display] \displaystyle \frac{2.829\times {{10}^{-9}}}{3.45\times {{10}^{-3}}}=8.2\times {{10}^{-7}}[/latex-display]Example
[latex] \displaystyle \frac{\left(1.37\times10^{4}\right)\left(9.85\times10^{6}\right)}{5.0\times10^{12}}[/latex]
Answer: Regroup the terms in the numerator according to the associative and commutative properties.
[latex] \displaystyle \frac{\left( 1.37\times 9.85 \right)\left( {{10}^{6}}\times {{10}^{4}} \right)}{5.0\times {{10}^{12}}}[/latex]
Multiply.[latex] \displaystyle \frac{13.4945\times {{10}^{10}}}{5.0\times {{10}^{12}}}[/latex]
Regroup using the associative property.[latex] \displaystyle \left( \frac{13.4945}{5.0} \right)\left( \frac{{{10}^{10}}}{{{10}^{12}}} \right)[/latex]
Divide the numbers.[latex] \displaystyle \left(2.6989\right)\left(\frac{10^{10}}{10^{12}}\right)[/latex]
Divide the powers of [latex]10[/latex] using the Quotient Rule—subtract the exponents.[latex] \displaystyle \begin{array}{c}\left(2.6989 \right)\left( {{10}^{10-12}} \right)\\2.6989\times {{10}^{-2}}\end{array}[/latex]
Answer
[latex-display] \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-display]example
Divide. Write answers in decimal form: [latex]{\Large\frac{9\times {10}^{3}}{3\times {10}^{-2}}}[/latex].Answer: Solution
| [latex]{\Large\frac{9\times {10}^{3}}{3\times {10}^{-2}}}[/latex] | |
| Separate the factors. | [latex]{\Large\frac{9}{3}}\times {\Large\frac{{10}^{3}}{{10}^{-2}}}[/latex] |
| Divide [latex]9[/latex] by [latex]3[/latex] and use the Quotient Property to divide [latex]{10}^{3}[/latex] by [latex]{10}^{-2}[/latex] . | [latex]3\times {10}^{5}[/latex] |
| Change to decimal form by moving the decimal five places right. | [latex]300,000[/latex] |
try it
[ohm_question]146319[/ohm_question] [ohm_question]2829[/ohm_question]Contribute!
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Examples: Dividing Numbers Written in Scientific Notation. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Examples: Multiplying Numbers Written in Scientific Notation. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
