Negative Exponents

Exponents are also called Powers or Indices

Let us first look at what an "exponent" is:

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 can be called "8 to the second power", "8 to the power 2"
or simply "8 squared"

 

Example: 53 = 5 × 5 × 5 = 125

In words: 53 can be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

In general:

an tells you to use a in a multiplication n times:   exponent definition

But those are positive exponents, what about something like:

8-2

That exponent is negative ... what does it mean?

Negative Exponents

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

Dividing is the inverse (opposite) of Multiplying.

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

Example: 8-1 = 1 ÷ 8 = 1/8 = 0.125

Or 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

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)

 

To change the sign (plus to minus, or minus to plus) of the exponent,
use the Reciprocal (i.e. 1/an)

 

So, what about 8-2 ?

Example: 8-2 = 1 ÷ 8 ÷ 8 = 1/82 = 1/64 = 0.015625

 

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

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.