+
2 Tutorials that teach The Product Property of Exponents
Take your pick:
The Product Property of Exponents

The Product Property of Exponents

Rating:
Rating
(0)
Description:

In this lesson, students will learn how to multiply exponential expressions by using the product property of exponents.

(more)
See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to over 2,000 colleges and universities.*

Begin Free Trial
No credit card required

25 Sophia partners guarantee credit transfer.

221 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 20 of Sophia’s online courses. More than 2,000 colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial
This tutorial covers multiplying monomials utilizing the product property of exponents, through the definition and discussion of:
  1. Monomials
  2. Multiplying Monomials/The Product Property of Exponents
  3. Rational Exponents


1. Monomials

Monomials are exponential expressions with non-negative integer exponents. There are three parts to a monomial: the base, the coefficient, and the exponent. The base is repeatedly multiplied by itself according to the exponent.

Monomial
An exponential expression with non-negative integer exponents

Consider the example of a monomial shown here:

File:1168-prod1.PNG

In this example, the base is x. The exponent is 3, and indicates how many times the base is used in repeated multiplication. The coefficient is 5, which is the number that acts as a multiplier to the base and the exponent.

It’s important to note that the exponent is only being applied to the base and not the coefficient.


2. Multiplying Monomials/The Product Property of Exponents

You can multiply monomials with and without coefficients. Consider this example of an expression without coefficients:

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

To simplify, you can expand the expression using repeated multiplication. Therefore, x squared becomes x times x, and x to the fourth becomes x times x times x times x. Multiplying the x terms together provides x to the sixth.

left parenthesis x times x right parenthesis left parenthesis x times x times x times x right parenthesis equals x to the power of 6

In the expression above, you can also determine the exponent of the product by adding the two exponents, 2 plus 4, together to provide x to the sixth. This illustrates the product property for exponential expressions, which works for the product of all monomials when the bases of the monomials are the same. In general, this states that x^a times x^b is equal to x^(a+b).

KEY FORMULA
left parenthesis x to the power of a right parenthesis left parenthesis x to the power of b right parenthesis equals x to the power of left parenthesis a plus b right parenthesis end exponent

This next example involves multiplying monomials with coefficients:

left parenthesis negative 3 x right parenthesis left parenthesis 5 x to the power of 6 right parenthesis

You can use the commutative property of multiplication, which allows you to group your coefficients and variables together. Multiplication is commutative because you can multiply numbers in any order. Therefore, grouping your coefficients and your x terms together gives you:

left parenthesis negative 3 times 5 right parenthesis left parenthesis x times x to the power of 6 right parenthesis

Multiplying -3 times 5 equals -15, and multiplying x times x^6 is the same as x^1 times x^6. Adding your exponents using the product property for exponents gives you x^7, providing your final solution of:

negative 15 x to the power of 7


3. Rational Exponents

Exponents are generally integers or fractions, but they may be any number, such as a decimal. A rational exponent is an exponent that can be represented as a fraction.

The product property applies to all types of exponents, including integers and fractions. Therefore, you add fractions when applying the product property of exponents to rational exponents. You can easily add fractions when they have common denominators:

  • First, write equivalent fractions with a common denominator.
  • Next, add the numerators, leaving the denominator unchanged.
  • Finally, reduce the fraction, if possible, by canceling out common factors in the numerator and denominator.

Suppose you want to simplify the following expression using the product property with rational exponents.

left parenthesis 2 x to the power of begin inline style 1 half end style end exponent right parenthesis left parenthesis 4 x to the power of 5 over 2 end exponent right parenthesis
Start by multiplying your coefficients, 2 times 4. Then, group your x terms together:
left parenthesis 2 times 4 right parenthesis left parenthesis x to the power of 1 half end exponent times x to the power of 5 over 2 end exponent right parenthesis

You need to add your exponents, meaning that you need to add the fractions together. Your denominators are already the same, so you can simply add your numerators together.

1 half plus 5 over 2 equals 6 over 2

Now, both your numerator and denominator have a common factor of 2, so you can cancel it out by dividing them both by 2, which gives you 3 over 1, or 3.

6 over 2 equals fraction numerator 6 divided by 2 over denominator 2 divided by 2 end fraction equals 3 over 1 equals 3

Bringing back in the product of your coefficients, your final expression becomes:

8 x cubed
Suppose you want to simplify the following expression:
left parenthesis negative x to the power of 3 over 4 end exponent right parenthesis left parenthesis 7 x to the power of 1 over 10 end exponent right parenthesis
Start by multiplying your coefficient. The negative in front of the x is the same as a coefficient of a negative one. Therefore, you have -1 times 7, which is -7. Next, group your x terms together to multiply them:
table attributes columnalign left end attributes row cell left parenthesis negative 1 times 7 right parenthesis left parenthesis x to the power of 3 over 4 end exponent times x to the power of 1 over 10 end exponent right parenthesis equals end cell row cell left parenthesis negative 7 right parenthesis left parenthesis x to the power of 3 over 4 end exponent times x to the power of 1 over 10 end exponent right parenthesis end cell end table
Remember, using the product property of exponents, when multiplying your x terms, you will add the exponents—which in this case are fractions—together. When adding your fractions together, notice that the denominators are not the same. Therefore, you need to think of the least common denominator.
The least common denominator of 4 and 10 is 20. To get a denominator of 20 in the first fraction, multiply by 5 in the denominator and the numerator. To get a denominator of 20 in the second fraction, multiply by 2 in the denominator and the numerator. Now that you have a common denominator, you can add your numerators, arriving at the fraction 17/20. This fraction is fully simplified because 17 and 20 have no other common factor than 1.
table attributes columnalign left end attributes row cell 3 over 4 plus 1 over 10 equals end cell row cell fraction numerator 3 times 5 over denominator 4 times 5 end fraction plus fraction numerator 1 times 2 over denominator 10 times 2 end fraction equals end cell row cell 15 over 20 plus 2 over 20 equals end cell row cell 17 over 20 end cell end table
Bringing back in the product of your coefficients, then, your final expression is:
negative 7 x to the power of 17 over 20 end exponent

Today you learned the definition of monomials and about the three parts of a monomial: the base, the coefficient, and the exponent. You also learned how to multiply monomials, with and without coefficients, using the product property of exponents, which states that the product of two monomials with the same base can be simplified by adding the original exponents. Finally, you learned about rational exponents and how to multiply monomials with rational exponents using the product property.

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

TERMS TO KNOW
  • KEY FORMULA

    (x^a)(x^b) = x^(a+b)

  • Monomial

    An exponential expression with non-negative integer exponents.