Power Of A Quotient: Property & Rule

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: What is an Imaginary Number?

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:02 Power of a Quotient
  • 1:26 How Does It Work?
  • 2:06 Examples
  • 3:47 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

In this lesson, learn about the power of a quotient property rule and how it's used in algebraic expressions. We will explain the rule, give some examples and provide a quiz for you at the end.

Power of a Quotient Property Rule

The Power of a Quotient rule is another way you can simplify an algebraic expression with exponents. Let's start by defining some terms as they relate to exponents. When you have a number or variable raised to a power, the number (or variable) is called the base, while the superscript number is called the exponent or power.

If there is more than one term in parentheses, with an exponent outside the parentheses then the exponent is distributed to every term in the parentheses.

For example, if you had this problem: (m/n)^5. Because of the parentheses, the exponent (5) should be distributed to each term. Therefore, (m/n)^5 = m^5/n^5.

Three conditions must be met in order for the Power of a Quotient rule to work.

  1. There must be two or more variables or constants that are being divided. In the above example, those are the m and n, but they could be any variable or constant.
  2. The result of the division problem must be raised to a power. In the above example, that is the 5.
  3. The variable in the denominator cannot be zero, and the numerator and exponent cannot both be zero. Keeping with the above example, this means that n does not equal zero. Both m and the exponent do not equal zero.

How Does It Work?

The Power of a Quotient rule can be proven by testing it using only numbers.


Using the Power of a Product rule, the solution is

4^3 / 2^3 = 64 / 8 = 8

Then, work the problem like a simple math problem.

(4/2)^3 = 2^3 = 8

No matter what two numbers and exponent you use, the answer reached mathematically will always equal the answer found when you use the Power of a Quotient rule to solve it. Therefore, using the Power of a Quotient rule will also work when the problem contains variables.


You can use the Power of a Quotient rule for simple, or more complex problems.

Example 1: Simplify (a/b)^7

Since both conditions are met (there are two variables being divided, and the result of that division problem is raised to a power), just distribute the exponent to both terms. (a/b)^7 = a^7/b^7

Example 2: Simplify (2x/y)^3

Distribute the 3, and simplify. 2^3 * x^3/y^3 = 8x^3/y^3

Example 3. Simplify. If the terms in the parentheses also have exponents, you can still simplify the problem. To distribute exponents, you multiply the exponents together. (3/a^2)^3

When you distribute the 3, you get the following: 3^3/a^6. Since 2 * 3 = 6, the exponent with the a will be a 6. The final answer is 27/a^6.

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account