In Exercises 59–72, simplify each expression using the products-t...

Question

In Exercises 59–72, simplify each expression using the products-to-powers rule.(5x³y⁻⁴)⁻²

Accepted Answer

Hey, everyone in this problem, we're asked to use the power rule to work out the given expression. The expression we're given is 12 X to the exponent seven Y to the exponent negative two, all to the exponent negative two. We're given four answer choices. Option A Y to the exponent four divided by 144 X to the exponent 14. Option B one divided by X to the exponent 14 Y to the exponent four. Option C negative Y to the exponent four divided by 144 X to the exponent 14. And option D negative one divided by X to the exponent 14 Y to the exponent four. We're gonna start by rewriting the expression we were given. So we have 12 x to the exponent seven Y to the exponent -2 all to the exponent negative too. Now when we have things multiplied together that are raised to an exponent. OK. That exponent applies to each of those terms. So we can write this as 12 to the exponent negative two multiplied by X to the exponent 7 to the exponent -2 multiplied by Y to the exponent -2 to the exponent negative too. Now, there's two things we want to deal with here on our first term, we have this negative exponent or call. If we have a negative exponent, say X to the exponent negative M, we can write this as one divided by X to the exponent positive M with our X and Y terms, we have an exponent raised to an exponent. How do we simplify that? Well, recall that if we have something, say A to the exponent M to the exponent and we can simplify this into a single exponent by multiplying the two exponents together. So we can write this as a to the exponent M multiplied by M. So apply both of these rules 12 to the exponent -2. It's going to be equal to one divided by 12 scripted and we move it to the denominator and the exponent becomes positive to the exponent seven to the exponent negative two. And we're gonna multiply the exponents and we get X to the exponent negative four, Y to the exponent negative two to the exponent negative two. Again, we multiply the exponent negative two multiplied by negative two gives us four. So we get Y to the exponent four. Now in our answer choices, we have all positive exponents. So we wanna get this X to the exponent negative 14 to be a positive exponent. So we're gonna apply the exact same rule that we used for 12 to the exponent negative two. Simplifying one divided by 12 squared. This is one divided by X to the exponent -14. We can write as one divided by X to the exponent A. We move it to the denominator. The exponent becomes positive and then we have multiplied by Y to the exponent or which we leave alone. There's no more simplifying to do with that one. Now, we can write this more simplified in the numerator, we have Y to the exponent four. And in the denominator, we have 144 multiplied by X to the exponent 14. And that is the simplified expression that we were asked to find and we have all of the exponents are positive and we can compare this with their answer choices. We found the correct answer was Y to the exponent four divided by 144 X to the exponent 14 which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 59–72, simplify each expression using the products-to-powers rule.(5x³y⁻⁴)⁻²

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