Power of a Product Rule for Exponents: A Few Quick Tips
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How to use the Power of a Product Property for Exponents
The Power of a Product Rule is another way to simplify exponential terms. There are a few things to consider when using the Power of a Product Rule to simplify exponents. First, you must have two terms being multiplied inside a set of parenthesis. Second, the two terms must also be being raised to an additional power.
In order to simplify using the Power of a Product Rule, you must distribute the power on the outside of the parenthesis to every term on the inside of the parenthesis. You will then multiply the exponent of each term with the power that was on 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
A Brief Power of a Product Rule Definition
The Power of a Product Rule is another approach to solve exponential terms. There are a couple of interesting points when using the Power of a Product Rule to solve exponents. To begin with, you should have two terms being raised inside a set. Second, the two terms should also be being raised to an extra power.
To use the Power of a Product Rule, you should apply the power to each term within the set. You will then multiply the power of each term with the power that was outside of the set.
If there is more than one term in set, you must distribute to each term in the set.
4 Steps for Solving any Power of a Product Example
- Power of a Product Rule occurs when there are multiple terms inside parenthesis and a power on the outside of the parenthesis.
- You must distribute the power of the outside of the parenthesis to the terms on the inside of the parenthesis.
- If there are many terms inside of the parenthesis, you must distribute to them all.
- To simplify, you multiply the powers inside the parenthesis with the power that was on the outside.
Power of a Product Property Practice Problems Quiz
Watch the Power of a Product Property Video Explanation
Watch our free video on how to solve Power of a Product Property. This video shows how to solve problems that are on our free Power of a Product Property worksheet that you can get by submitting your email above.
Watch the free Power of a Product video on YouTube here: Power of a Product
This video is about the power of product rule for exponents. You can get this power of a product rule worksheet for free by clicking on the link in the description below.
Here we are at the first problem for the power of a product rule. The first problem gives us 3 times 10 raised to the second power, now this 2 means that we are squaring the quantity 3 times 10. We could rewrite 3 times 10 as 3 times 10 times 3 times 10. We have our quantity 3 times 10 squared. Now we have the quantity 3 times 10 times itself one time because we’re squaring it if you were to simplify this you would rewrite it as the threes together and with the tens together. We would have 3 times 3 times 10 times 10 and then to simplify this 3 times 3 you could write as 3 squared and then 10 times 10 you could write as 10 squared. This will be your solution and is a short explanation of the power of a product definition.
Now there is a shortcut for product of a power rule that will allow you to skip the steps in the middle, if you remember back to the power of a power property anytime you have an exponent on the outside of the parenthesis you have to distribute it to everything on the inside of the parenthesis. When we have our term here 3 times 10 squared we could easily take the 2 and distribute it to the 3 and the 10 that’s on the inside of the parenthesis. We would rewrite it as 3 to the second power times 10 to the 2nd power, and in this case that is your solution you can see that both of these answers are exactly the same, and that’s how you know that the shortcut will work.
Let’s move on to a little bit more difficult power of a product examples and if we look at number six it gives us x squared times three to the seventh the quantity raised to the fifth power. Now we already know that we can take this five here and we can distribute it to everything that’s on the inside of the parenthesis. We will distribute the five to the eight squared and the exponent of the 5 to the 3 to the seventh power. When we do that we’re going to write a squared in parentheses, that’s our first term and it’s being distributed with the 5. The exponent of the 5 has to go to the a squared then we have three to the seventh and then the exponent of the 5 is also being applied to the 3 to the seventh. We took the 8 squared we wrote it here with the exponent of 5 on the outside and then we took the 3 to the seventh and we rewrote it as 3 to the seventh raised to the fifth power. You will remember that when we have a parenthesis in between two exponents that means you’re going to multiply those exponents together. That parenthesis means 2 times 5 or this parenthesis here means 7 times 5 so we keep the base in this case 8 to the 2nd raised to the fifth power. You would do 8 keep the base and then 2 times 5 for the exponents and then 2 times 5 is 10, and then you do the same thing with the 3, you keep the 3 as the base and then for the exponent you do 7 times 5 which is of course 35. That’s your answer.
The last example we’re working on is going to have variables instead of numbers as the bases. Now when we have variables nothing about the rule will change, so we will still take the exponent on the outside and distribute it to everything on the inside. In the case of a number 9 we have X to the fifth times y to the sixth times Z to the seventh the whole quantity raised to the second power. We will take this 2 and the two will get distributed to everything on the inside of the parenthesis, so 2 goes to the X to the fifth Y to the sixth and the Z to the seventh. When this is distributed we’re gonna rewrite X to the fifth which now has the exponent of two on above it times the quantity or the term y to the sixth raised to the second power times the term Z to the seventh also being raised to the second power. We can go ahead and multiply the exponents because we know that the parentheses in between the exponents means to multiply. We’ll have to keep the base of X and then we’ll do 5 times 2 which is 10. Then we’ll keep the base of Y and then we’ll do 6 times 2 which is 12. And then we’ll keep the base of Z and we will do 7 times 2 which is 14. Our final answer is X to the 10th times y to the 12th times Z to the 14th. You can try all the practice problems by downloading the free product of powers worksheet above.
Power of a Product Rule
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