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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³√32x⁹y¹⁷

Accepted Answer

Welcome back. I am so glad you're here. We're asked to simplify the following radical expression by factoring, assume that C is greater than zero and D is greater than zero. The expression we're given is the cubed rut of 54 C raised to the 12th D raised to the 14th. Our answer choices are answer choice A two C raised to the fourth D multiplied by the crate of 3d squared. Answer choice B two C race to the fourth D race to the fourth multiplied by the cub of 3d squared. Answer choice C three C raced to the fourth D multiplied by the crate of two D squared and answer choice D three C race to the fourth D race to the fourth multiplied by the cub of two D squared. All right, we look at our radical expression and we note that the index of our radical is three. That means that there must be three of any factor in order for it to come out of the radical. So we need to look at our 54, our C raise to the 12th and our D race to the 14th to see if they contain three of any factor. Let's look at 54 1st, we can put 54 in a factor tree. 54 is nine, multiplied by 69, is three, multiplied by three and six is three multiplied by two. And look here, we have three of the factor three or we could write that as three cubed and then that is still multiplied by two, we can expand our radical. So we still have the cubed rut of. But now writing instead of writing 54 we have three multiplied by two. Now we can expand C raised to the 12th power. C race to the 12th. Power is the same as C raise to the fourth multiplied by C raise to the fourth multiplied by C race to the fourth. Because we recall from previous lessons that when we have like bases, those are the CS and we multiply them together, we will be adding their exponents. So four plus four plus four equals 12 and we could rewrite C raise to the fourth multiplied by itself three times as C raise to the fourth cubed. So putting that back underneath a radical, we have three cubed multiplied by two, multiplied by C race to the 4th cubed. Now we can take a look at D raised to the 14th. So we have D raise to the 14th, which could be expanded as D raise to the fourth multiplied by D raise to the fourth multiplied by D raise to the fourth multiplied by D squared. Because if we take four plus four plus four plus two, that will equal 14, and we could combine the D raise to the fourth multiplied by itself three times as D raise to the 4th cubed. And that is still multiplied by D squared. So now we can put this back inside of our cubed rot. So we have three cubed multiplied by two multiplied by C race to the fourth cubed multiplied by D raised to the 4th cubed. And that is multiplied by D squared. And now anything that has three of it as a factor may come outside of the radical, that's going to be our three cubed C race to the fourth cubed D race to the fourth cubed. And when they come outside of that cubed rut, they lose their cube. So coming out, it will just be three C ray to the fourth D race to the fourth. And that's multiplied by the cub rot of. And what's left behind is the two multiplied by D squared. And that is as simplified as this radical expression gets. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

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. ______ ³√32x⁹y¹⁷

Key Concepts

Factoring

Radical Expressions

Properties of Exponents

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