Exponents

Definition

The exponent of a number says how many times to use the number in a multiplication.

In 42  the “2” says to use 4 twice in a multiplication,

so

42 = 4 × 4 = 16

In words:

42 could be called “4 to the power 2” or “4 to the second power”, or simply “4 squared”

Exponents are also called Powers or Indices.

Some more examples:

In NumbersIn Words
33 = 3 × 3 × 3 = 27“3 to the third power”, “3 to the power 3”,or simply “3 cubed”
54 = 5 × 5 × 5 × 5 = 625“5 to the fourth power”,”5 to the power 4″, or simply “5 to the 4th”

Exponents make it easier to write and use many multiplications

Example

76 is easier to write and read than 7 × 7 × 7 × 7 × 7 × 7

We can multiply any number by itself as many times as we want using exponents.

In General

an tells you to multiply a by itself, so there are n of those a‘s:

Exponent

If the Exponent is 1 or 0?

ExponentAnswerExample
1number itself81 = 8
0160 = 1

What about 00 ?

It could be either 1 or 0, and so people say it is “indeterminate“.

Negative Exponents

What could be the opposite of multiplying?

Dividing!

A negative exponent means how many times to divide one by the number.

Example

4-1 = 1 ÷ 4 = 0.25

You can have many divides

Examples

10-2 = 1 ÷ 10 ÷ 10 = 0.01

That can be done in another way:

10-2 could also be calculated like:

1 ÷ (10 × 10) = 1/102 = 1/100 = 0.01

Negative Exponent = Flip the Positive Exponent

That last example showed an easier way to handle negative exponents:

  1. Calculate the positive exponent (an)
  2. Then take the Reciprocal (1/an)

More Examples:

Negative ExponentReciprocal of Positive ExponentAnswer
5-21/521/25 = 0.04
2-31/231/8 = 0.125
(-10)-31/(-10)31/(-1000) = -0.001

Makes Sense

Start with “1” and then multiply or divide as many times as the exponent says, then you will get the right answer.

Example

etc…n-2n-1Number (n)n1n2etc…
 …2-2=1÷2÷22-2=0.252-1 = 1÷22-1 = 0.5221 = 1×221 = 222=1×2×222=4 …
 …4-2=1÷4÷44-2=0.06254-1 = 1÷44-1 = 0.25441 = 1×441 = 442=1×4×442=16 …
 …5-2=1÷5÷55-2=0.045-1 = 1÷55-1 = 0.2551 = 1×551 = 552=1×5×552=25 …

Another Way of Writing Exponent

Sometimes people use the ^ symbol (above the 6 button on your keyboard), as it is easy to type.

Example

3^2 is the same as 32

3^2 = 3 × 3 = 9

About Grouping

To avoid confusion, use parentheses () in cases like this:

With ( )Without ( )
(-7)2 = (-7) × (-7) = 49-72 = -(72) = – (7 × 7) = -49
(ab)2 = ab × abab2 = a × (b)2 = a × b × b

Learn More

Negative Exponents

Fractional Exponents

Squares and Square Roots

Cube Numbers and Cube Roots

Algebra Index

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