Negative Exponents

Exponents are also called Powers or Indices

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"

So, in general:

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

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

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.