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 $a\times10^{n}$ ). Then multiply the powers of ten by adding the exponents.

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’t, you convert it.

Let’s look at some examples.

Example

$\left(3\times10^{8}\right)\left(6.8\times10^{-13}\right)$

Show Solution

Regroup using the commutative and associative properties.

$\left(3\times6.8\right)\left(10^{8}\times10^{-13}\right)$

Multiply the coefficients.

$\left(20.4\right)\left(10^{8}\times10^{-13}\right)$

Multiply the powers of $10$ using the Product Rule. Add the exponents.

$20.4\times10^{-5}$

Convert $20.4$ into scientific notation by moving the decimal point one place to the left and multiplying by $10^{1}$ .

$\left(2.04\times10^{1}\right)\times10^{-5}$

Group the powers of $10$ using the associative property of multiplication.

$2.04\times\left(10^{1}\times10^{-5}\right)$

Multiply using the Product Rule—add the exponents.

$2.04\times10^{1+\left(-5\right)}$

Answer

$\left(3\times10^{8}\right)\left(6.8\times10^{-13}\right)=2.04\times10^{-4}$

Example

$\left(8.2\times10^{6}\right)\left(1.5\times10^{-3}\right)\left(1.9\times10^{-7}\right)$

Show Solution

Regroup using the commutative and associative properties.

$\left(8.2\times1.5\times1.9\right)\left(10^{6}\times10^{-3}\times10^{-7}\right)$

Multiply the numbers.

$\left(23.37\right)\left(10^{6}\times10^{-3}\times10^{-7}\right)$

Multiply the powers of $10$ using the Product Rule—add the exponents.

$23.37\times10^{-4}$

Convert $23.37$ into scientific notation by moving the decimal point one place to the left and multiplying by $10^{1}$ .

$\left(2.337\times10^{1}\right)\times10^{-4}$

Group the powers of $10$ using the associative property of multiplication.

$2.337\times\left(10^{1}\times10^{-4}\right)$

Multiply using the Product Rule and add the exponents.

$2.337\times10^{1+\left(-4\right)}$

Answer

$\left(8.2\times10^{6}\right)\left(1.5\times10^{-3}\right)\left(1.9\times10^{-7}\right)=2.337\times10^{-3}$

example

Multiply. Write answers in decimal form: $\left(4\times {10}^{5}\right)\left(2\times {10}^{-7}\right)$ .

Show Solution

Solution

	$\left(4\times {10}^{5}\right)\left(2\times {10}^{-7}\right)$
Use the Commutative Property to rearrange the factors.	$4\cdot 2\cdot {10}^{5}\cdot {10}^{-7}$
Multiply $4$ by $2$ and use the Product Property to multiply ${10}^{5}$ by ${10}^{-7}$ .	$8\times {10}^{-2}$
Change to decimal form by moving the decimal two places left.	${\Large\frac{8}{100}} = 0.08$

try it

In the following video you will see an example of how to multiply tow numbers that are written in scientific notation.

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 $10$ (the a in $a\times10^{n}$ . 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

$\displaystyle \frac{2.829\times 1{{0}^{-9}}}{3.45\times 1{{0}^{-3}}}$

Show Solution

Regroup using the associative property.

$\displaystyle \left( \frac{2.829}{3.45} \right)\left( \frac{{{10}^{-9}}}{{{10}^{-3}}} \right)$

Divide the coefficients.

$\displaystyle \left(0.82\right)\left( \frac{{{10}^{-9}}}{{{10}^{-3}}} \right)$

Divide the powers of $10$ using the Quotient Rule. Subtract the exponents.

$\begin{array}{l}0.82\times10^{-9-\left(-3\right)}\\0.82\times10^{-6}\end{array}$

Convert $0.82$ into scientific notation by moving the decimal point one place to the right and multiplying by $10^{-1}$ .

$\left(8.2\times10^{-1}\right)\times10^{-6}$

Group the powers of $10$ together using the associative property.

$8.2\times\left(10^{-1}\times10^{-6}\right)$

Multiply the powers of $10$ using the Product Rule—add the exponents.

$8.2\times10^{-1+\left(-6\right)}$

Answer

$\displaystyle \frac{2.829\times {{10}^{-9}}}{3.45\times {{10}^{-3}}}=8.2\times {{10}^{-7}}$

Example

$\displaystyle \frac{\left(1.37\times10^{4}\right)\left(9.85\times10^{6}\right)}{5.0\times10^{12}}$

Show Solution

Regroup the terms in the numerator according to the associative and commutative properties.

$\displaystyle \frac{\left( 1.37\times 9.85 \right)\left( {{10}^{6}}\times {{10}^{4}} \right)}{5.0\times {{10}^{12}}}$

Multiply.

$\displaystyle \frac{13.4945\times {{10}^{10}}}{5.0\times {{10}^{12}}}$

Regroup using the associative property.

$\displaystyle \left( \frac{13.4945}{5.0} \right)\left( \frac{{{10}^{10}}}{{{10}^{12}}} \right)$

Divide the numbers.

$\displaystyle \left(2.6989\right)\left(\frac{10^{10}}{10^{12}}\right)$

Divide the powers of $10$ using the Quotient Rule—subtract the exponents.

$\displaystyle \begin{array}{c}\left(2.6989 \right)\left( {{10}^{10-12}} \right)\\2.6989\times {{10}^{-2}}\end{array}$

Answer

$\displaystyle \frac{\left( 1.37\times {{10}^{4}} \right)\left( 9.85\times {{10}^{6}} \right)}{5.0\times {{10}^{12}}}=2.6989\times {{10}^{-2}}$

example

Divide. Write answers in decimal form: ${\Large\frac{9\times {10}^{3}}{3\times {10}^{-2}}}$ .

Show Solution

Solution

	${\Large\frac{9\times {10}^{3}}{3\times {10}^{-2}}}$
Separate the factors.	${\Large\frac{9}{3}}\times {\Large\frac{{10}^{3}}{{10}^{-2}}}$
Divide $9$ by $3$ and use the Quotient Property to divide ${10}^{3}$ by ${10}^{-2}$ .	$3\times {10}^{5}$
Change to decimal form by moving the decimal five places right.	$300,000$

try it

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.

The following video is a mini-lesson on how to convert decimals to scientific notation, and back to a decimal. Additionally, you will see more examples of how to multiply and divide numbers given in scientific notation.

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Module 1: Exponents and Radicals