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 8⁶ and it will mean the same thing.

When reading 8⁶, 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 = 12¹⁰

In 12¹⁰, 12 is called the base and 10 is called the exponent.

Other examples of exponents:

5³ = 5 × 5 × 5

9⁴ = 9 × 9 × 9 × 9

7² = 7 × 7

6⁶ = 6 × 6 × 6 × 6 × 6 × 6 × 6

2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

In one of the examples above, it says 7² = 7 × 7

What would you say 7¹ is equal to?

For 7², you wrote down 7 twice.

Therefore, for 7¹, you will write down 7 once and of course there is no need to have a multiplication sign.

7¹ = 7

Common pitfalls to avoid when working with exponents:

What is -2⁶ 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) = -(2⁶) = - 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 = + × + × + × 2⁶

Note that + × + = +

(-2)⁶ = +2⁶

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 = + × + × + × - × 2⁷

+ × + × + × - × 2⁷ = + × - × 2⁷

Note that + × - = -

+ × - × 2⁷ = -2⁷

Important observations about exponents:

• -aⁿ 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.

• -aⁿ is not always equal to (-a)ⁿ

• -aⁿ 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:

Exponents quiz to check your understanding of this lesson.