# 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:**
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 our example 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 not 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}
+ × + = +

+ 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}
+ × - = -

+ × - × 2

^{7} = -2

^{7}
**Observation:**
-a

^{n} is always negative

(-a)

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

-a

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

^{n}
-a

^{n} is 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 to use parentheses as shown below:

The above is also equal to

8
/
27

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

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