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Multiplying Terms using Exponent Properties

Author: Sophia

what's covered
This tutorial covers multiplying terms using exponent properties, through the exploration of:

Table of Contents

1. Components of Exponential Expressions

In review, there are three components of exponential expressions: the base, the coefficient, and the exponent. In the term below, the a is the base, the 9 is the coefficient, and the 3 is the exponent or power.


2. Product and Power Properties of Exponents

You may recall the product property of exponents, shown below, which states that when you multiply exponential terms with the same base, you can add their exponents to simplify.

formula to know
Product Property of Exponents
open parentheses x to the power of m close parentheses open parentheses x to the power of n close parentheses equals x to the power of open parentheses m plus n close parentheses end exponent

EXAMPLE

As shown below, when you multiply these two exponential terms, since they have the same base, you can add simply add their exponents.

open parentheses x squared close parentheses open parentheses x to the power of 7 close parentheses equals x to the power of open parentheses 2 plus 7 close parentheses end exponent equals x to the power of 9

The second property, the power property of exponents, states that when taking the power of an exponential expression, the exponents are multiplied, as shown below:

formula to know
Power Property of Exponents
open parentheses x to the power of m close parentheses to the power of n equals x to the power of m times n end exponent

EXAMPLE

Consider the expression:

open parentheses x to the power of 4 close parentheses cubed

It is the same as:

open parentheses x to the power of 4 close parentheses open parentheses x to the power of 4 close parentheses open parentheses x to the power of 4 close parentheses

You can use the product property of exponents to add the fours together—4 plus 4 plus 4—which equals 12.

open parentheses x to the power of 4 close parentheses cubed equals open parentheses x to the power of 4 close parentheses open parentheses x to the power of 4 close parentheses open parentheses x to the power of 4 close parentheses equals table attributes columnalign left end attributes row cell x to the power of open parentheses 4 plus 4 plus 4 close parentheses end exponent equals x to the power of 12 end cell end table

We could also use the power property to multiply the exponents, which in this case, would be 4 times 3.

open parentheses x to the power of 4 close parentheses cubed equals x to the power of open parentheses 4 times 3 close parentheses end exponent equals x to the power of 12

Either way, the expression is equal to x to the power of 12.

key concept
When using the power property, you sometimes need to multiply fractional exponents. This involves multiplying across the numerators and denominators of the fractions. Fractions are reduced by canceling common factors of the numerator and denominator.


3. Multiplying Terms Using Exponent Properties

Suppose you want to simplify the expression:

open parentheses x to the power of 1 half end exponent close parentheses to the power of 2 over 3 end exponent

The power property of exponents can be used to multiply the two fractional exponents to get a single exponent.

x to the power of open parentheses 1 half times 2 over 3 close parentheses end exponent

Remember, when you multiply fractions, you multiply straight across—numerator by numerator and denominator by denominator. Therefore, you have the following, which can be simplified because 2 is a common factor of both 2 and 6. Dividing the numerator and denominator by 2 gives us 1/3.

x to the power of open parentheses 1 half times 2 over 3 close parentheses end exponent equals x to the power of 2 over 6 end exponent equals x to the power of fraction numerator 2 divided by 2 over denominator 6 divided by 2 end fraction end exponent equals x to the power of 1 third end exponent

Now, use what you’ve learned about the product and power properties of exponents to simplify a more complicated expression.

try it
Consider the following expression:

open parentheses x to the power of 3 over 5 end exponent times x to the power of 7 over 10 end exponent close parentheses to the power of 1 half end exponent
Simplify this expression.
Start by simplifying in your parentheses. You have two exponential terms multiplying together, and they both have a base of x; therefore, you can use the product property of exponents, and add your exponents together. In this case, your exponents are fractions, so to add them, you need a common denominator. The least common denominator of 5 and 10 is 10. Multiply your first fraction by 2 in the denominator and the numerator, and leave your second fraction unchanged. Now that your denominator is the same, you can add your numerators.

3 over 5 plus 7 over 10 equals fraction numerator 3 times 2 over denominator 5 times 2 end fraction plus 7 over 10 equals 6 over 10 plus 7 over 10 equals 13 over 10

Your expression becomes:

open parentheses x to the power of 3 over 5 end exponent times x to the power of 7 over 10 end exponent close parentheses to the power of 1 half end exponent equals open parentheses x to the power of 13 over 10 end exponent close parentheses to the power of 1 half end exponent

You can now use the power property to multiply 13/10 and 1/2, multiplying straight across, numerator by numerator and denominator by denominator. You don’t need to simplify the fraction because 13 and 20 do not have any common factors other than 1:

x to the power of open parentheses 13 over 10 times 1 half close parentheses end exponent equals x to the power of 13 over 20 end exponent

summary
Today you reviewed the three components of an exponential expression: coefficients, bases, and exponents. You also learned about two properties of exponents: the product property of exponents, which states that when multiplying exponential terms together with the same base, the exponents are added; and the power property of exponents, which states that when taking the power of an exponential expression, the exponents are multiplied. Finally, you practiced multiplying terms using exponent properties.

Source: This work is adapted from Sophia author Colleen Atakpu.

Formulas to Know
Power Property of Exponents

open parentheses x to the power of m close parentheses to the power of n equals x to the power of m times n end exponent

Product Property of Exponents

open parentheses x to the power of m close parentheses open parentheses x to the power of n close parentheses equals x to the power of left parenthesis m plus n right parenthesis end exponent