## 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

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

### Example

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

### example

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

### 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

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

### Example

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

### example

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

### 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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