Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

5.1.2 Using Product and Quotient Properties for Exponents

Algebra 15.1.2 Using Product and Quotient Properties for Exponents

Search for key terms or text.

Activity

Product Property for Exponents

If a a is a real number and m m and n n are integers, then

a m · a n = a m + n a m · a n = a m + n

To multiply with like bases, add the exponents.

Complete problems 1 - 6 using the product property for exponents.

1. d 2 · d 3 d 2 · d 3

2. c 4 · c 6 c 4 · c 6

3. k 3 · k 5 k 3 · k 5

4. ( v ) 2 · ( v ) 1 ( v ) 2 · ( v ) 1

5. ( m ) 4 · ( m ) 4 ( m ) 4 · ( m ) 4

6. ( p ) 6 · ( p ) 8 ( p ) 6 · ( p ) 8

Product Quotient for Exponents

If a a is a real number, a 0 a 0 , and m m and n n  are integers, then

a m a n = a m n a m a n = a m n , m > n m > n and a m a n = 1 a n m a m a n = 1 a n m , n > m n > m

To divide with like bases, subtract the exponents.

Complete problems 7 - 12 using the quotient property for exponents.

7. y 5 y 3 y 5 y 3

8. r 4 r r 4 r

9. q 9 q 2 q 9 q 2

10. ( u ) 5 ( u ) 2 ( u ) 5 ( u ) 2

11. ( j ) 6 ( j ) 5 ( j ) 6 ( j ) 5

12. ( z ) 8 ( z ) 6 ( z ) 8 ( z ) 6

Self Check

Which of the following equations is correct?
  1. ( u ) 8 ( u ) 2 = 2 ( u ) 4
  2. y 6 y 3 = y 2
  3. x 2 x 4 = x 8
  4. y 5 y 3 = y 8

Additional Resources

Using Product and Quotient Properties for Exponents

When we combine like terms by adding and subtracting, we 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.

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

What does x 2 · x 3 x 2 · x 3 mean?

An equation shows two x multiplied (2 factors) and three x multiplied (3 factors), separated by a dot, combining to make five x multiplied (5 factors).

Notice that 5 is the sum of the exponents, 2 and 3. We see x 2 · x 3 x 2 · x 3 is x 2 + 3 x 2 + 3 or x 5 x 5 .

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

Product Property for Exponents

If a a is a real number and m m and n n are integers, then

a m · a n = a m + n a m · a n = a m + n

To multiply with like bases, add the exponents.

Example 1

4 3 · 4 4 = 4 3 + 4 = 4 7 4 3 · 4 4 = 4 3 + 4 = 4 7

Example 2

12 2 · 12 6 = 12 2 + 6 = 12 8 12 2 · 12 6 = 12 2 + 6 = 12 8

Example 3

y 7 · 4 8 = y 7 + 8 = y 15 y 7 · 4 8 = y 7 + 8 = y 15

Example 4

( 2 b ) 10 · ( 2 b ) 3 = ( 2 b ) 10 + 3 = ( 2 b ) 13 ( 2 b ) 10 · ( 2 b ) 3 = ( 2 b ) 10 + 3 = ( 2 b ) 13

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

1. Consider x 5 x 2 x 5 x 2

Step 1 - What does it mean?

x · x · x · x · x x · x x · x · x · x · x x · x

Step 2 - Use the Equivalent Fractions Property.

x   ·   x   ·   x   ·   x   ·   x x   ·   x x   ·   x   ·   x   ·   x   ·   x x   ·   x

Step 3 - Simplify.

x 3 x 3

2. Consider x 2 x 3 x 2 x 3

Step 1 - What does it mean?

x · x x · x · x x · x x · x · x

Step 2 - Use the Equivalent Fractions Property.

x   ·   x   ·   1 x   ·   x   ·   x x   ·   x   ·   1 x   ·   x   ·   x

Step 3 - Simplify.

1 x 1 x

Notice that in each case, the bases were the same and we subtracted exponents. We see x 5 x 2 x 5 x 2 is x 5 2 x 5 2 or x 3 x 3 . We see x 2 x 3 x 2 x 3 is 1 x 1 x . When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, so we need a 1 in the numerator. x x   = 1 x x   =1 . This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If a a is a real number, a 0 a 0 , and m m and n n are integers, then

a m a n = a m n a m a n = a m n , m > n m > n and a m a n = 1 a n m a m a n = 1 a n m , n > m n > m

Example 5

7 5 ÷ 7 3 = 7 5 3 = 7 2 7 5 ÷ 7 3 = 7 5 3 = 7 2

Example 6

k 9 ÷ k 2 = k 9 2 = k 7 k 9 ÷ k 2 = k 9 2 = k 7

Example 7

j 10 j 7 = j 10 7 = k 3 j 10 j 7 = j 10 7 = k 3

Try it

Try It: Using Product and Quotient Properties for Exponents

Simplify each expression.

  1. 12 2 · 12 5 12 2 · 12 5
  2. 20 14 ÷ 20 4 20 14 ÷ 20 4
  3. h 4 · h 7 h 4 · h 7
  4. ( s ) 2 · ( s ) 2 ( s ) 2 · ( s ) 2
  5. b 7 b 5 b 7 b 5
  6. ( y ) 6 ( y ) 3 ( y ) 6 ( y ) 3
  7. x 3 x 5 x 3 x 5

Video: Understanding Product and Quotient Properties for Exponents

Watch the following video to learn more about applying the Product and Quotient Properties of Exponents.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.