Multiplying with exponents, or exponentiated numbers, requires a different approach than multiplying regular numbers. This is because exponents are single-term exponential expressions.

### What are Exponents?

There are two main components to a single-term exponential expression: the base and the power. In the expression three is the base and four is the power. This is read as “three raised to the power of four” or “three raised to the fourth power”.

A number being raised to a power means that number is being multiplied by itself that number of times. For 3^{4}, three is being multiplied by itself four times. That is:

3 x 3 x 3 x 3 =81.

### Multiplying with Exponents

When multiplying exponents, there are three cases to consider. Exponents can have the same base and different powers, different bases and the same power, or different powers and different bases. Each of these is handled differently.

The first case is multiplying with exponents with different powers and bases. Solve each expression and multiply their answers.

*a ^{n}* x

*b*

^{m}*a*=

^{n}*c*

*b*=

^{m}*d*

*a*=

^{n}x b^{m}*c x d*

When multiplying exponents with different bases and the same power, multiply the bases and leave the powers alone. Then, solve the final expression.

*a ^{n}* x

*b*

^{n}*= (a*x

*b)*

^{n}Finally, multiplying with exponents with the same base and different powers. Add their powers and then solve.

*a ^{n}* x

*a*

^{m}*= a*

^{n+m}The following examples display each of these cases step-by-step.

#### Example 1: Different Powers and Bases

6^{3} x 4^{5}

Step 1: Solve each exponential expression.

6^{3 }= 6 x 6 x 6 = 216

4^{5 }= 4 x 4 x 4 x 4 x 4 =1,024

Step 2: Multiply the corresponding solutions.

216 x 1,024 = 221,184

#### Example 2: Different Bases and the Same power

8^{5} x 5^{4}

Step 1: Multiply the bases.

(8 x 5)^{4}= 40^{4}

Step 2: Solve the final expression.

40^{4 }= 2,560,000

#### Example 3: Same Base and Different Power

7^{3} x 7^{6}

Step 1: Add the powers.

7^{3+6} = 7^{9}

Step 2: Solve the final expression.

7^{9 }= 40,353,607

#### Example 4

Introduced by John Wallis, fractional exponents are another way to represent the roots of a number.

4^{5/2 }x 4^{2/3}

Step 1: Identify the case and its required method.

This problem has two expressions with the same bases and different powers.

Step 2: Add the powers.

4^{5/2 + 2/3} x 4^{15/6 + 4/6 }= 4^{21/6 }= 4^{7/2}

Step 3: Solve the final expression.

4^{7/2}= 128

#### Example 5

6^{3} x 5^{2}

Step 1: Identify the case and its required method.

This problem has different bases and powers.

Step 2: Solve each expression.

6^{3 }= 216

5^{2 }= 25

Step 3: Multiply the corresponding solutions.

216 x 25 = 5,400

#### Example 6

4^{5} x 3^{5}

Step 1: Identify the case and its required method.

This problem has two different bases and the same power.

Step 2: Multiply the bases.

(4 x 3)^{5} = 12^{5}

Step 3: Solve the final expression.

12^{5 }= 248,832

#### Example 7

2^{7} x 2^{7}

Step 1: Identify the case and its required method.

This problem has the same base and power. You may choose either of the applicable methods to solve it.

Step 2: Add the powers OR multiply the bases.

2^{(7+7)} = 2^{14}

OR

(2 x 2)^{7} = 4^{7}

Step 3: Solve the final expression.

2^{4 }= 16,384

OR

4^{7 }= 16,384

As you can see, multiplying with exponents isn’t as difficult as it seems. Just keep in mind the required methods. Then, when multiplying exponents, you’ll be done in 1-2-3!

Author: Khaiylah Bustamante