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Exponents

Exponents are also called Powers or Indices

8 to the Power 2

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

In this example: 82 = 8 × 8 = 64

  • In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

 

Some more examples:

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

You can multiply any number by itself as many times as you want using exponents.

In General

So, in general:

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

 

Other Way of Writing It

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

Example: 2^4 is the same as 24

  • 2^4 = 2 × 2 × 2 × 2 = 16

 

Negative Exponents

Negative? What could be the opposite of multiplying? Dividing!

A negative exponent means how many times to divide 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

negative-exponent

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

  • Calculate the positive exponent (an)
  • Then take the Reciprocal (i.e. 1/an)

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

What if the Exponent is 1, or 0?

1   If the exponent is 1, then you just have the number itself (example 91 = 9)
     
0   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
  .. etc..  
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
  .. etc..  

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