# Using Exponents in Algebra

You might like to read the page on Exponents first.

## Whole Number Exponents

The exponent "n" in a^{n} says **how many times** to use a in a multiplication:

### Example: **5**^{3} = 5 × 5 × 5 = 125

^{3}= 5 × 5 × 5 = 125

- The "3" says to use 5 three times in a multiplication
- In words: 5
^{3}could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

## Negative Exponents

A negative exponent a^{-n} is equal to 1/a^{n} (1 divided by the positive exponent)

### Example:

**5 ^{-3} =**

**1/5**

^{3}= 1/125 = 0.008Also ... by changing the signs of the exponents we get:

A positive exponent a^{n} is equal to 1/a^{-n} (1 divided by the negative exponent)

So, we can move an expression between the top and bottom (numerator and denominator) of a fraction when we also change the sign of the exponent.

Example: **x ^{-1} =**

**1/x**(a simple reciprocal)

Example: **5 ^{-3} =**

**1/5**

^{3}= 1/125 = 0.008## Positive and Negative Together

Here is an example with positive and negative exponents:

### Example: 4^{3}2^{-5}9^{-1}3^{2}

We can put the values with negative exponents at the bottom (remembering to make the exponents positive):

\frac{4^{3}3^{2}}{2^{5}9}

Let's simplify it!

3^{2} is 9:

\frac{4^{3}9}{2^{5}9}

The 9s cancel out:

\frac{4^{3}}{2^{5}}

**4 ^{3}** is 4 × 4 × 4 = 64, and

**2**is 2 × 2 × 2 × 2 × 2 = 32:

^{5}\frac{64}{32}

Which simplifies to:

2

Done!