# Power Of A Quotient: Property & Rule

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• 0:02 Power of a Quotient
• 1:26 How Does It Work?
• 2:06 Examples
• 3:47 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
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.

(4/2)^3

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.

## Examples

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.

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