What is the name for the number that tells you how many times the base of a power is multiplied by itself?

An exponent tells you how many times a number (or other quantity) is multiplied by itself.

The Oxford English Dictionary defines an exponent as "a symbol denoting the number of times a particular quantity is to be taken as a factor to produce the power indicated."

Table of Contents Show

Example: 53 = 5 × 5 × 5 = 125
Example: 24 = 2 × 2 × 2 × 2 = 16
Another Way of Writing It
Negative Exponents
Negative? Flip the Positive!
What if the Exponent is 1, or 0?
It All Makes Sense
Be Careful About Grouping

It is easier to understand exponents with an example. Imagine we wanted to multiply 3 by itself 2 times. We would write 3 with a small 2 written to the right and above it: 32.

A power is the product of multiplying a number by itself. 32 is equal to 3 multiplied by itself 2 times, which equals 9.
3 is called the base. It is the number that is multiplying itself.
2 is called the exponent. It tells you how many times the base is multiplying itself.

Some real examples of exponents are given below.

The power below has an exponent of 3:
The power below has an exponent of 4:
The exponent can also be a letter. The power below has an exponent of n:

Exponents can also be a negative number. A negative exponent tells you how many times to divide 1 by the number. For example, 3−2 means divide 1 by 3, 2 times:

A negative exponent means put the base under 1 (or turn it upside-down if the base is a fraction), and make the exponent positive.

Read more about how to find a negative exponent An exponent can be a fraction. A fractional exponent means finding a root of a number. For example, 3½ means the square root of 3:

An exponent can also be called a power or an index (plural indices). How do you say 32? You could say:

3 to the power of 2.
The second power of 2.
3 to the 2.
In the special case where the exponent is 2, we can say 3 "squared".

Note: In the special case where the exponent is 3, as in 33, we can say 3 "cubed". There are several laws of exponents, which are very useful in algebra. We use these laws when we are multiplying and dividing bases with exponents. They also help us understand negative and fractional exponents.
Read more about the laws of exponents If a number or a letter does not have an exponent, imagine it has an invisible exponent of 1:

Lesson Objectives

Demonstrate an understanding of multiplication with whole numbers
Demonstrate an understanding of multiplication by powers of 10 (trailing zeros)
Learn how to write the repeated multiplication of the same whole number using exponents
Learn how to evaluate an exponential expression
Learn how to quickly evaluate 10 raised to a whole number exponent

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

In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64

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

Note: Exponents are also called Powers or Indices.

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

Try here:

algebra/images/exponent-calc.js

So in general:

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

Another Way of Writing It

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

Example: 2^4 is the same as 24

Negative Exponents

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

So we divide by the number each time, which is the same as multiplying by 1number

Example: 8-1 = 18 = 0.125

We can continue on like this:

Example: 5-3 = 15 × 15 × 15 = 0.008

But it is often easier to do it this way:

5-3 could also be calculated like:

15 × 5 × 5 = 153 = 1125 = 0.008

Negative? Flip the Positive!

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
(-2)-3	=	1 / (-2)3	=	1/(-8) = -0.125

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".