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

### Example

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

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

### Example

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

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.