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.

(-3x⁻²)⁻³

Accepted Answer

Hey, everyone in this problem, we're asked to use the power rule to work out the given expression. We have negative five, Z to the exponent negative six all to the exponent negative two. We're given four answer choices. Option A Z to the exponent negative 12 divided by 25. Option B negative Z to the exponent 12 divided by 25. Option C negative Z to the exponent negative 12 divided by 25. And option D the Z to the exponent 12 divided by 25. We're gonna start by rewriting what we were given. So we have negative five Z to the exponent negative six up to the exponent negative too. Now when we have things multiplied together and they're raised to an exponent that exponent applies to both of those terms. So we can write this as negative five to the exponent negative two multiplied by Z to the exponent negative six to the exponent negative two. OK. That exponent of negative two has to apply to both of those. Now we have two things to deal with here. We have negative five to a negative exponent. How do we deal with the negative exponent Well recall our rule. If we have a to the exponent negative M, we can write this as one divided by A to the exponent positive M. So if we have a negative exponent, we can take that base and the exponent move them to the denominator and the exponent becomes positive. Now, we also have Z to the exponent -6 to the exponent -2. How do we deal with an exponent to an exponent? Will we use our power rule? So if we have a to the exponent M and raised to the power of N, you can write this as a to the exponent M multiplied by N. So an exponent raised to a power, we can simplify it by multiplying those two exponents together. So applying these rules to our problem, we have negative five to the exponent negative two. We can move this to the denominator to make that exponent positive. And we have one divided by negative five squared. And then we have Z to the exponent negative six to the exponent negative two. We can simplify that using our power rule, we multiply the exponents negative six multiplied by negative two. And we get Z to the exponent 12. Now simplifying our denominator in the numerator, we have Z to the exponent 12. In the denominator, we have negative five squared which gives us a value of 25. OK. We have a negative to an even power. So that's a positive value. So we get 25 in our denominator. And that is the simplified expression that we were trying to find. OK. We found that the correct answer is Z to the exponent divided by 25 which corresponds with answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 59–72, simplify each expression using the products-to-powers rule. (-3x⁻²)⁻³

Key Concepts

Exponents and Powers

Negative Exponents

Products-to-Powers Rule

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