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.