Exponents

Posted on Updated on Exponents are also called Powers or Indices. The exponent of a number says how many times to use the number in a multiplication.

Example: 53 = 5 × 5 × 5 = 125

In words: 53 could be called “5 to the third power”, “5 to the power 3” or simply “5 cubed”

Example: 24 = 2 × 2 × 2 × 2 = 16

In words: 24 could be called “2 to the fourth power” or “2 to the power 4” or simply “2 to the 4th”

Exponents make it easier to write and use many multiplications

Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

In general:

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

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

Example: 8-1 = 1 ÷ 8 = 0.125

You can have many divides:

Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

5-3 could also be calculated like:   1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

In General

More Examples:

Negative Exponent Reciprocal of Positive Exponent Answer
4-2 = 1 / 42 = 1/16 = 0.0625
10-3 = 1 / 103 = 1/1,000 = 0.001
(-2)-3 = 1 / (-2)3 = 1/(-8) = -0.125

What if the Exponent is 1, or 0?

 If the exponent is 1, then you just have the number itself (example 91 = 9) If the exponent is 0, then you get 1 (example 90 = 1) But what about 00 ? It could be either 1 or 0, and so people say it is “indeterminate”.

It All Makes Sense

My favorite method is to start with “1” and then multiply or divide as many times as the exponent says,

then you will get the right answer, for example:

Example: Powers of 5 52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.

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

 (-2)2 = (-2) × (-2) = 4 –22 = -(22) = – (2 × 2) = -4
 (ab)2 = ab × ab ab2 = a × (b)2 = a × b × b