# How to Solve the Power of a Quotient Rule for Exponents

Get the free Power of a Quotient worksheet and other resources for teaching & understanding solving Power of a Quotient

**Here’s how to use the Power of a Quotient Property**

The **Power of a Quotient Rule** is another way to simplify exponential terms. There are a few things to consider when using the **Power of a Quotient Rule** to simplify exponents. First, you must have at least two terms being divided inside a set of parenthesis. Second, the terms must also be being raised to an additional power that is outside of the parenthesis.

In order to simplify using the **Power of a Quotient Rule**, you must distribute the power on the outside of the parenthesis to every term on the inside of the parenthesis. If there are like terms, you should simplify them first. Then you will then multiply the exponent of each term with the power that was distributed from the outside of the parenthesis.

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

**Common Core Standard:** 8.EE.A.1

**Related Topics:**Product Rule, Quotient Rule, Power of a Power Rule, Power of a Product Rule, Negative Exponents

**The Concise Power of a Quotient Rule Definition**

The Power of a Quotient Rule is another approach to solving exponential terms. There are a couple of interesting points when using the Power of a Quotient Property to simplify examples. To begin with, you should have something like two terms being divide inside a set. Second, the terms should be being raised to an extra power that is outside of the set.

To use the Power of a Quotient Property, you should apply the exponent outside of the set to each term within the set. Always simplify Like Terms first. After they are simplified, you will then apply the exponent outside of the set.

In the event that there is more than one term in set, with an exponent outside the set, it is applied to each term in the set.

**4 Steps for solving this Example of Power of a Quotient**

- Distribute the exponent on the outside of the parenthesis to everything on the inside of the parenthesis.
- After you distribute, you multiply the exponents.
- When multiplying, make sure you check the exponents for negatives and include any negatives when multiplying.
- Reduce any Like Terms if there are any.

**Power of a Quotient Rule Practice Problems Quiz**

**Watch the Power of a Quotient Rule Property Video Explanation**

Watch our free video on how to use **Power of a Quotient Property**. This video shows how to solve problems that are on our free **Power of a Quotient Property **worksheet that you can get by submitting your email above.

**Watch the free Power of a Quotient video on YouTube here: Power of a Quotient**

**Video Transcript:**

This video is about the power of a quotient rule for exponents. You can get the quotient of powers worksheet used in this video for free by clicking on the link in the description below.

The first problem on the power of quotient worksheet gives us the term 3 to the fourth divided by five to the ninth inside parentheses raised to the second power. You should already know that anytime you have an exponent on the outside of the parenthesis you have to distribute it to all the exponents on the inside of the parenthesis. Now in order to show this to you we’re going to take each term and we’re going to rewrite it with the exponent on the outside of the parenthesis. Our top term is 3 to the fourth it is being raised to the second power and that’s over five to the ninth which is also being raised to the second power. Now you could take this 3 to the fourth and you could write it out 3 to the fourth times 3 to the fourth and you can do the same thing for five to the ninth times 5 to the 9.

We just took each term squared and we rewrote it and we wrote it out then you could simplify it. Simplifying this you do 4 plus 4 which would be 3 to the 8th all over and then 5 to the 9th times 5 to the ninth would be 5 to the 9 plus 9 which is 18. That’s going to be our answer but there is a shortcut.

If we jump back to our first step and we have 3 or 4 squared over 5 to the 9 squared we already know that anytime you have a parenthesis in between two exponents that means you’re going to multiply them. This would be 3 to the 4 times 2 which would be 3 to the 8th and then 5 to the 9th times 2 which would be 5 to the 18th. You’ll notice that both answers are identical. We can go ahead and skip this middle step if we want to.

The next problem we’re going to work on on the quotient of powers property worksheet gives us five to the ninth over five to the fifth raised to the second power. Now this problem has one additional step that the first problem did not. If you look at this problem we already know that we have to distribute everything on the outside of parentheses to everything on the inside of the parenthesis. We will take five to the ninth and we will square it all over five to the fifth, that is also being squared. When you simplify this you’re going to multiply the exponents together. And in this case five to the nine times two would be five to the 18th and then five to the fifth times two would be five to the tenth.

Typically we would be done at this step except we still have like terms that can be reduced. We have five to the 18th divided by five to the tenth and we know that when we divide exponential terms you are going to subtract them. 5 to the 18th minus 10 which would be 5 to the 8th, and that’s going to be our final answer.

The last problem we’re going to go over on our power of a quotient rule worksheet is number 9. And that is X to the fourth Y to the 9th over x squared Y to the fifth and then the whole term is being raised to the second power. Now we already know we have to take this 2 and you have to distribute it to everything on the inside of the parenthesis. That includes all of the X’s and all of the Y’s. We will rewrite our problem x to the fourth squared Y to the ninth squared all over x squared squared times y to the fifth squared. We already know that if you have a parenthesis in between two exponents that means to multiply. We’re going to multiply all of these exponents together when we do that we will get X to the 8th Y to the 18th all over X to the fourth Y to the 10th.

Then the last step is that we have to simplify we have an X on top and X on bottom and a Y on top in a Y bottom. When we simplify this we will subtract the exponents so we will do X to the 8th minus 4 which is 4 and then y to the 18th minus 10 which is 8. And that’s our final answer for this quotient of powers property problem.

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