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"

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"

And 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 this notation.

So, in general:

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

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

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)

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?

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)

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.