In Exercises 39–60, simplify by factoring. Assume that all variab...

Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers.
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√x⁶y⁷

Accepted Answer

Welcome back. I am so glad you're here. We are asked to simplify the following radical expression by factoring assume that M is greater than zero and N is greater than zero. The expression we're given is the square root of M raised to the 12th power N raised to the fifth power. Our answer choices are answer choice am raise to the sixth power N squared square rot N answer choice B M squared N raised to the sixth power square root N answer choice C M raise to the sixth power N squared square root M and answer choice D M squared N raise to the sixth power square root M. All right. The first thing we do when simplifying radical expressions is we take a look and see what our index is. And when it's not written, it is a two, that two tells us that we have to have two of any factor in order for it to come outside of the radical. Looking at the variables inside the radical, we have M raised to the 12th power and N raised to the fifth power. We want to group those, we want to expand them. And break them down and group them so that they come in twos so that they can escape the radical. We can expand M rays to the 12th power and set that equal to M ray to the sixth power multiplied by M raise to the sixth power because we recall from previous lessons when we have like bases, those are the MS, we are and they're being multiplied together, we are going to add their exponents six plus six equals 12. But we can also rewrite this as M M M rays to the sixth power multiplied by M rays to the sixth power is the same as M rays to the sixth power squared. And so we can expand this radical, we're taking the square root of and instead of M raised to the 12th power, we have M raised to the sixth power squared. Now expanding our N rays to the fifth power, that is the same as N squared multiplied by N squared multiplied by N because adding those exponents two plus two plus the unwritten one for just N gives us five. And so we can focus on this N squared multiplied by N squared. That's the same as N squared squared still multiplied by N. But that gives us something that's squared for underneath our radical. So back in our radical, we have M raced to the sixth power squared. Now that's multiplied by N squared squared and that is multiplied by N. Now we look for things that have a power of two that can come outside of the radical. That's going to be our M raced to the sixth and N squared. When they come outside of the radical, they lose that squared. So it will just be M race to the sixth power N squared. That's multiplied by the square root of what's left inside is just N and that is as simplified as this one gets, We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

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In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. ___ √x⁶y⁷