Ulearngo

Simplifying Expressions Using the Product Property For Exponents

Simplifying Expressions Using the Product Property For Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

We’ll derive the properties of exponents by looking for patterns in several examples.

First, we will look at an example that leads to the Product Property.

x squared times x cubed.
What does this mean?
How many factors altogether?		 x times x, multiplied by x times x. x times x has two factors. x times x times x has three factors. 2 plus 3 is five factors.
So, we have x to the fifth power.
Notice that 5 is the sum of the exponents, 2 and 3. x squared times x cubed is x to the power of 2 plus 3, or x to the fifth power.

We write:

\(\begin{array}{c}\hfill {x}^{2}·{x}^{3}\hfill \\ \hfill {x}^{2+3}\hfill \\ \hfill {x}^{5}\hfill \end{array}\)

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If \(a\) is a real number, and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are counting numbers, then

\({a}^{m}·{a}^{n}={a}^{m+n}\)

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

\(\begin{array}{ccc}\hfill {2}^{2}·{2}^{3}& \stackrel{?}{=}\hfill & {2}^{2+3}\hfill \\ \hfill 4·8& \stackrel{?}{=}\hfill & {2}^{5}\hfill \\ \hfill 32& =\hfill & 32\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}\)

Example

Simplify: \({y}^{5}·{y}^{6}.\)

Solution

y to the fifth power times y to the sixth power.
Use the product property, am · an = am+n. y to the power of 5 plus 6.
Simplify. y to the eleventh power.

Example

Simplify:

  1. \({2}^{5}·{2}^{9}\)
  2. \(3·{3}^{4}.\)

Solution

  1. 2 to the fifth power times 2 to the ninth power.
    Use the product property, am · an = am+n. 2 to the power of 5 plus 9.
    Simplify. 2 to the 14th power.
  2. 3 to the fifth power times 3 to the fourth power.
    Use the product property, am · an = am+n. 3 to the power of 5 plus 4.
    Simplify. 3 to the ninth power.

Example

Simplify:

  1. \({a}^{7}·a\)
  2. \({x}^{27}·{x}^{13}.\)

Solution

  1. a to the seventh power times a.
    Rewrite, a = a1. a to the seventh power times a to the first power.
    Use the product property, am · an = am+n. a to the power of 7 plus 1.
    Simplify. a to the eighth power.
  2. x to the twenty-seventh power times x to the thirteenth power.
    Notice, the bases are the same, so add the exponents. x to the power of 27 plus 13.
    Simplify. x to the fortieth power.

We can extend the Product Property for Exponents to more than two factors.

Example

Simplify: \({d}^{4}·{d}^{5}·{d}^{2}.\)

Solution

d to the fourth power times d to the fifth power times d squared.
Add the exponents, since bases are the same. d to the power of 4 plus 5 plus 2.
Simplify. d to the eleventh power.

This lesson is part of:

Polynomials II

View Full Tutorial

Track Your Learning Progress

Sign in to unlock unlimited practice exams, tutorial practice quizzes, personalized weak area practice, AI study assistance with Lexi, and detailed performance analytics.