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 34, 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.
an x bm
an = c
bm = d
an x bm = 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.
an x bn = (a x b)n
Finally, multiplying with exponents with the same base and different powers. Add their powers and then solve.
an x am = an+m
The following examples display each of these cases step-by-step.
Example 1: Different Powers and Bases
63 x 45
Step 1: Solve each exponential expression.
63 = 6 x 6 x 6 = 216
45 = 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
85 x 54
Step 1: Multiply the bases.
(8 x 5)4= 404
Step 2: Solve the final expression.
404 = 2,560,000
Example 3: Same Base and Different Power
73 x 76
Step 1: Add the powers.
73+6 = 79
Step 2: Solve the final expression.
79 = 40,353,607
Example 4
Introduced by John Wallis, fractional exponents are another way to represent the roots of a number.
45/2 x 42/3
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.
45/2 + 2/3 x 415/6 + 4/6 = 421/6 = 47/2
Step 3: Solve the final expression.
47/2= 128
Example 5
63 x 52
Step 1: Identify the case and its required method.
This problem has different bases and powers.
Step 2: Solve each expression.
63 = 216
52 = 25
Step 3: Multiply the corresponding solutions.
216 x 25 = 5,400
Example 6
45 x 35
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 = 125
Step 3: Solve the final expression.
125 = 248,832
Example 7
27 x 27
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) = 214
OR
(2 x 2)7 = 47
Step 3: Solve the final expression.
24 = 16,384
OR
47 = 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