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2 Tutorials that teach Multiplying Terms using Exponent Properties
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Multiplying Terms using Exponent Properties

Multiplying Terms using Exponent Properties

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In this lesson, students will learn how to multiply terms by using the product and power properties of exponents.

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Tutorial
This tutorial covers multiplying terms using exponent properties, through the exploration of:
  1. Components of Exponential Expressions
  2. Product and Power Properties of Exponents
  3. Multiplying Terms Using Exponent Properties


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.

File:1172-exp1.PNG


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.

KEY FORMULA
left parenthesis x to the power of m right parenthesis left parenthesis x to the power of n right parenthesis equals x to the power of left parenthesis m plus n right parenthesis end exponent

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

left parenthesis x squared right parenthesis left parenthesis x to the power of 7 right parenthesis equals x to the power of left parenthesis 2 plus 7 right parenthesis 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:

KEY FORMULA
left parenthesis x to the power of m right parenthesis to the power of n equals x to the power of m times n end exponent

Consider the expression:

left parenthesis x to the power of 4 right parenthesis cubed

It is the same as:

left parenthesis x to the power of 4 right parenthesis left parenthesis x to the power of 4 right parenthesis left parenthesis x to the power of 4 right parenthesis

You can use the product property of exponents to add the fours together—4 + 4 + 4—which is the same as 4 times 3—which equals 12. Therefore, the expression is equal to x^12.

table attributes columnalign left end attributes row cell x to the power of left parenthesis 4 plus 4 plus 4 right parenthesis end exponent equals end cell row cell x to the power of 4 times 3 end exponent equals end cell row cell x to the power of 12 end cell end table
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:

left parenthesis x to the power of 1 half end exponent right parenthesis 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 left parenthesis 1 half times 2 over 3 right parenthesis 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.

table attributes columnalign left end attributes row cell x to the power of 2 over 6 end exponent equals end cell row cell x to the power of fraction numerator 2 divided by 2 over denominator 6 divided by 2 end fraction end exponent equals end cell row cell x to the power of 1 third end exponent end cell end table
Now, use what you’ve learned about the product and power properties of exponents to simplify a more complicated expression:
left parenthesis x to the power of 3 over 5 end exponent times x to the power of 7 over 10 end exponent right parenthesis to the power of 1 half end exponent
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.
table attributes columnalign left end attributes row cell 3 over 5 plus 7 over 10 equals end cell row cell fraction numerator 3 times 2 over denominator 5 times 2 end fraction plus 7 over 10 equals end cell row cell 6 over 10 plus 7 over 10 equals end cell row cell 13 over 10 end cell end table
Your expression becomes:
left parenthesis x to the power of 13 over 10 end exponent right parenthesis 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:
table attributes columnalign left end attributes row cell x to the power of left parenthesis 13 over 10 times 1 half right parenthesis end exponent equals end cell row cell x to the power of 13 over 20 end exponent end cell end table

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.

TERMS TO KNOW
  • KEY FORMULA

    (x^m)^n = x^(mn)

  • KEY FORMULA

    (x^m)(x^n) = x^(m+n)