# Exponents

An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 2^{3}) means:

**2 x 2 x 2 = 8**.

2^{3} is not the same as 2 x 3 = 6.

Remember that a number raised to the power of 1 is itself. For example,

**a ^{1} = a**

**5 ^{1} = 5**.

There are some special cases:

**1. a ^{0} = 1**

When an exponent is zero, as in 6^{0}, the expression is always equal to 1.

**a ^{0} = 1**

**6 ^{0} = 1**

**14,356 ^{0} = 1**

**2. a ^{-m} = 1 / a^{m}**

When an exponent is a negative number, the result is always a fraction. Fractions consist of a numerator over a denominator. In this instance, the numerator is always 1. To find the denominator, pretend that the negative exponent is positive, and raise the number to that power, like this:

**a ^{-m} = 1 / a^{m}**

**6 ^{-3} = 1 / 6^{3}**

You can have a variable to a given power, such as a^{3}, which would mean a x a x a. You can also have a number to a variable power, such as 2^{m}, which would mean 2 multiplied by itself m times. We will deal with that in a little while.

First let’s look at how to work with variables to a given power, such as a^{3}.

There are five rules for working with exponents:

1. a^{m} * a^{n} = a^{(m+n)}

2. (a * b)^{n} = a^{n} * b^{n}

3. (a^{m})^{n} = a^{(m * n)}

4. a^{m} / a^{n} = a^{(m-n)}

5. (a/b)^{n} = a^{n} / b^{n}

Let’s look at each of these in detail.

1. **a ^{m} * a^{n} = a^{(m+n)}** says that when you take a number, a, multiplied by itself m times, and multiply that by the same number a multiplied by itself n times, it’s the same as taking that number a and raising it to a power equal to the sum of m + n.

Here’s an example where

**a = 3**

m = 4

n = 5

**a ^{m} * a^{n} = a^{(m+n)}**

**3 ^{4} * 3^{5} = 3^{(4+5)} = 3^{9} = 19,683**

2. **(a * b) ^{n} = a^{n} * b^{n}** says that when you multiply two numbers, and then multiply that product by itself n times, it’s the same as multiplying the first number by itself n times and multiplying that by the second number multiplied by itself n times.

Let’s work out an example where

**a = 3**

b = 6

n = 5

**(a * b) ^{n} = a^{n} * b^{n}**

**(3 * 6) ^{5} = 3^{5} * 6^{5}**

**18 ^{5} = 3^{5} * 6^{5} = 243 * 7,776 = 1,889,568**

3. **(a ^{m})^{n} = a^{(m * n) }**says that when you take a number, a , and multiply it by itself m times, then multiply that product by itself n times, it’s the same as multiplying the number a by itself m * n times.

Let’s work out an example where

**a = 3**

**m = 4**

**n = 5**

**(a ^{m})^{n} = a^{(m * n)}**

**(3 ^{4})^{5} = 3^{(4 * 5)} = 320 = 3,486,784,401**

4. **a ^{m} / a^{n} = a^{(m-n)}** says that when you take a number, a, and multiply it by itself m times, then divide that product by a multiplied by itself n times, it’s the same as a multiplied by itself m-n times.

Here’s an example where

**a = 3
m = 4
n = 5**

**a ^{m} / a^{n} = a^{(m-n)}**

**3 ^{4} / 3^{5} = 3^{(4-5)} = 3^{-1} (Remember how to raise a number to a negative exponent.)**

**3 ^{4} / 3^{5} = 1 / 3^{1} = 1/3**

5. **(a/b) ^{n} = a^{n} / b^{n}** says that when you divide a number, a by another number, b, and then multiply that quotient by itself n times, it is the same as multiplying the number by itself n times and then dividing that product by the number b multiplied by itself n times.

Let’s work out an example where

**a = 3
b = 6
n = 5**

**(a/b) ^{n} = a^{n} / b^{n}**

**(3/6) ^{5} = 3^{5} / 6^{5}**

Remember 3/6 can be reduced to 1/2. So we have:

**(1/2) ^{5} = 243 / 7,776 = 0.03125**

Understanding exponents will prepare you to use logarithms.

**Video tutorials**

- https://www.khanacademy.org/math/algebra-basics/core-algebra-exponent-expressions/core-algebra-exponent-properties/v/exponent-properties-involving-products
- https://www.khanacademy.org/math/algebra-basics/core-algebra-exponent-expressions/core-algebra-exponent-properties/v/products-and-exponents-raised-to-an-exponent-properties
- https://www.khanacademy.org/math/algebra-basics/core-algebra-exponent-expressions/core-algebra-exponent-properties/v/exponent-properties-involving-quotients

**Example of exponent rules**