Multiply and Divide Numbers in Scientific Notation

Learning Outcome

Multiply and divide numbers expressed in scientific notation

Multiplying and Dividing 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 are not 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 is not, you convert it.

Let us look at some examples.

Example

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

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, so 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, so add the exponents.

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

The answer is $2.04\times10^{-4}$

Example

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

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, so 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, so add the exponents.

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

The answer is $2.337\times10^{-3}$

In the following video, you will see an example of how to multiply two numbers that are written 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 are not 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 is not already in scientific notation, you convert it, and then you are done.

Let us look at some examples.

Example

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

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, so 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, so add the exponents.

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

The answer is $8.2\times {{10}^{-7}}$

Example

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

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, so subtract the exponents.

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

The answer is $2.6989\times {{10}^{-2}}$

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.

Module 7: Exponents