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Multiplication with Exponents

by Ron Kurtus (revised 8 July 2019)

When you multiply exponential expressions, there are some simple rules to follow. If they have the same base, you simply add the exponents.

Note: The base of the exponential expression xy is x and the exponent is y.

This is also true for numbers and variables with different bases but with the same exponent. You can apply the rules when other numbers are included.

This rule does not apply when the numbers or variables have different bases and different exponents.

Questions you may have include:

This lesson will answer those questions.



Multiplying exponents with the same base

When you multiply two variables or numbers that have the same base, you simply add the exponents.

(xa)*(xb) = xa+b

Thus x3*x4 = x3+4 = x7.

Proof: 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

Demonstration with numbers

A demonstration of that rule is seen when you multiply 73 times 72. The result is:

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

7*7*7*7*7 = 75

Instead of writing out the numbers, you can simply add the exponents:

73*72= 73+2 = 75

Likewise, 23*25*22 = 23+5+2 = 210.

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

Different bases but same exponent

When you multiply two variables or numbers or with different bases but with the same exponent, you can simply multiply the bases and use the same exponent. For example:

(xa)*(ya) = (xy)a

Also:

(x3)*(y3) = xxx*yyy = (xy)3

Likewise, with numbers:

32*42= (3*4)2 = 122 = 144

Including other numbers

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

(12*75)*(2*73)

Rearrange the numbers:

(12*2)*(75*73)

Then add the exponents:

24*78

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) = 7y2x8

(a3*b3)*(b6*a5) = a8b9

When rule does not apply

When you multiply expressions with different bases and different exponents, there is no rule to simplify the process.

For example, suppose you want to multiply 23*52.

You can see that 23 = 8 and 52 = 25. Thus 8*25 = 200. But, if you tried (2*5)3+2, you would get 105, which is incorrect.

Summary

When you multiply two numbers or variables with the same base, you simply add the exponents. When you multiply expressions with the same exponent but different bases, you multiply the bases and use the same exponent.

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.

When you multiply expressions with different bases and different exponents, there is no rule to simplify the process.


Always do your best


Resources and references

Ron Kurtus' Credentials

Websites

Exponents: Basic Rules - PurpleMath.com

Exponent Rules - RapidTables.com

Laws of Exponents - MathisFun.com

Exponents Calculator - CalculatorSoup.com

Algebra Resources

Books

Top-rated books on Algebra


Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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