# Exponents

Exponents can make your math problems a lot easier to handle. Simply put, it is a shortcut for multiplying numbers over and over again.

Look at the following multiplication problem:

8 × 8 × 8 × 8 × 8 × 8

Instead of multiplying 8 six times by itself, we can just write 8

^{6} and it will mean the same thing.

When reading 8

^{6}, we say eight to the sixth power or eight to the power of six.

In a similar way,
12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 = 12

^{10}
In 12

^{10}, 12 is called the base and 10 is called the exponent.

## Other examples of exponents:

5

^{3} = 5 × 5 × 5

9

^{4} = 9 × 9 × 9 × 9

7

^{2} = 7 × 7

6

^{6} = 6 × 6 × 6 × 6 × 6 × 6 × 6

2

^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

**In general**,

y^{n} = y × y × y × y ×...× y × y (n times)

In one of the examples above, it says 7

^{2} = 7 × 7

What would you say 7

^{1} is equal to?

For 7

^{2}, you wrote down 7 twice.

Therefore, for 7

^{1}, you will write down 7 once and of course there is no need to have a multiplication sign.

7

^{1} = 7

## Common pitfalls to avoid when working with exponents:

What is -2

^{6} equals to?

Is it equal to -2 × -2 × -2 × -2 × -2 × -2 ?

Or is it equal to -(2 × 2 × 2 × 2 × 2 × 2) ?

It is equal to -(2 × 2 × 2 × 2 × 2 × 2) = -(2

^{6}) = - 64

However, (-2)

^{6} is a different story.

(-2)

^{6} = (-2) × (-2) × (-2) × (-2) × (-2) × (-2) = - × - × - × - × - × - × 2 × 2 × 2 × 2 × 2 × 2

Notice that - × - = +

So, - × - × - × - × - × - × 2 × 2 × 2 × 2 × 2 × 2 = + × + × + × 2

^{6}
Note that + × + = +

(-2)

^{6} = +2

^{6}
In (-2)

^{6}, the exponent is even. Change it to any odd number and the answer will be negative.

(-2)

^{7} = (-2) × (-2) × (-2) × (-2) × (-2) × (-2)× (-2) = - × - × - × - × - × - × - × 2 × 2 × 2 × 2 × 2 × 2 × 2

- × - × - × - × - × - × - × 2 × 2 × 2 × 2 × 2 × 2 × 2 = + × + × + × - × 2

^{7}
+ × + × + × - × 2

^{7} = + × - × 2

^{7}
Note that + × - = -

+ × - × 2

^{7} = -2

^{7}
## Important observations about exponents:

- -a
^{n} is always negative.

- (-a)
^{n} is either negative or positive. It is positive if n is an even number. It is negative if n is an odd number.

- -a
^{n} is not always equal to (-a)^{n}

- -a
^{n} **may** equal to (-a)^{n} only when n is an odd number.

**What if the base is a fraction?**
When the base is a fraction it is common to use parentheses as shown below:

The above is also equal to

8
/
27

## Exponents quiz to check your understanding of this lesson.