Home > Pre-Algebra > Exponents > Powers of Ten

## Powers of Ten

#### Introduction

Powers of ten are a way of using exponents to help explain very large (or very small) numbers. A power of 10 is the number 10 followed by an exponent. For example, $10^{2}$ = 100.

If you wanted to explain the distance from the Earth to the Sun in kilometers, you can use a power of 10 to make the number simpler.

The actual distance is 150,000,000 kilometers. However, we can use exponents to write this as $1.5 imes 10^{8}$ kilometers. By using the power of 10, we can eliminate the need to write multiple zeros at the end and yet still show the size of the number.

## Lesson

One way to think about powers of 10 is that the number of the exponent just tells us how many spaces to move the decimal point to the right. Take the following number:

$7.2 imes 10^{3}$

The $^{3}$ tells use to move the decimal point three spaces to the right. If we write 7.2 as 7.2000 we can move the decimal point three spaces to the right and it becomes 7200.0, or 7200.

Therefore, $7.2 imes 10^{3} = 7200$

We can do this with very big numbers as well:

$8.5 imes 10^{16}$

We need to move the decimal point 16 places to the right, giving us:

### Negative powers

We can also use powers of 10 to denote very small numbers. The trick here is to look out for negative exponents, which look like this.

$5.4 imes 10^{-3}$

To calculate this number, we need to move the decimal point to the left not the right.

So, if we rewrite $5.4 imes 10^{-3}$ as 00005.4, and then move the decimal point three places to the left, we get 0.0054.

We can also do this process with very small numbers:

$1.7 imes 10^{-12}$

Rewrite as: 0000000000001.7

Move the decimal point 12 spaces to the left to get 0.0000000000017

## Examples

$10{,}000$
$\left(10\right)^{3}$
$10 * 10 * 10$
$\left(10\right)^{3} = 1000$
$\left(10\right)^{-3}$
Rewrite with a positive exponent by taking the reciprocal of the base
$\dfrac{1}{\left(10\right)^{3}}$
$\dfrac{1}{1000}$