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Multiplication with Exponents

by Ron Kurtus (10 January 2008)

When you multiply two numbers or variables with the same base, you simply add the exponents. This is true for both numbers and variables. When you include other numbers in the multiplication, you simply break it up into several multiplications.

Questions you may have are:

• How do you multiply numbers raised to a power?
• How do you multiply variables raised to a power?
• How do you include other numbers when multiplying?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Multiplying exponential numbers

When you multiply two numbers or variables with the same base, you add the exponents.

A demonstration or verification of that rule is seen when you multiply 7*7*7 times 7*7. The result is:

(7*7*7)*(7*7) =

7*7*7*7*7 = 7⁵

But 7*7*7 = 7³ and 7*7 = 7². Thus, 7³*7²= 7³⁺² = 7⁵.

You can see that when you multiply numbers raised to a power, you add their exponents.

Multiplying variables

When you multiply two variables with the same base, you add the exponents. Thus x³*x⁴ = x³⁺⁴ = x⁷. This can be proved, since x³ = x*x*x and x⁴ = x*x*x*x, then

(x*x*x)*(x*x*x*x) =

(x*x*x)*(x*x*x*x) = x⁷

Also, when both the base and exponents are variables, (x^a)*(x^b) = x^a+b.

Note that you are multiplying with the same base numbers raised to powers. If the base numbers are different, this rule does not apply. Situations of multiplying 7³*4² or (x⁶)*(y⁶) cannot be simplified.

Including other numbers

If you have exponential numbers that are multiplied by other numbers, you can easily do the arithmetic. For example,

(12*10⁵)*(2*10³) =

(12*2)*(10⁵*10³) =

24*10⁸ =

2.4*10*10⁸ = 24*10⁹

Note that you change 24*10⁸ to 2.4*10⁹ because the standard convention is that the first number in raising 10 to a power should be between 1 and 10. Since 24 is more than 10, you can divide 24 by 10 and multiply 10⁸ by 10 to get 2.4*10⁹.

The other numbers or variables can also be exponentials. Some examples include:

(3³*5²)*(5³*3³) = (3³⁺³)*(5²⁺³) = 3⁶*5⁵

(7*x³)*(y²*x⁵) = 7*y²*x⁸

(a³*b³)*(b⁶*a⁵) = a⁸*b⁹

Summary

When you multiply two numbers or variables with the same base, you simply add the exponents. This is true for both numbers and variables. For example, 2³*2⁷= 2¹⁰ and c³*c⁴ = c⁷. When you include other numbers or variables in the multiplication, you simply break it up into several multiplications, such as (x*10⁵)*(x*10³) =x²*10⁸.

Answers to Readers' Questions

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Mini-quiz to check your understanding

1. What is 5²*3² in exponential notation?

2. What is x²*x^a in exponential notation?

3. What is (a^x*b^y)*(a^z*b^w) equal to?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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