In Exercises 29–44, simplify using the quotient rule.
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⁵√6...

Question

In Exercises 29–44, simplify using the quotient rule.
_______
⁵√64x¹⁴/y¹⁵

Accepted Answer

Welcome back. I am so glad you're here. We're asked to use the quotient rule for the simplification of the fifth wretch Of a fraction. In the numerator, we have 625 p raised to the 24th power. And in the denominator, we have Q raised to the 25th power. Our answer choices are answer choice. A P raised to the 4th power multiplied by the fifth of 625 P ray to the fourth power Divided by Q. Rays to the 5th power. Answer choice B P race to the fourth power multiplied by the fifth rut of 625 P raised to the fifth power Divided by Q Race to the 4th Power. Answer choice C five P rays to the fourth power multiplied by The 5th of P race to the 4th power Divided by Q. Rays to the 5th power and answer choice D P raises to the fourth power multiplied by the fifth rut of 625 P raised to the fourth power Divided by five Q Rays to the 5th power. All right. So the quotient rule tells us that when we are taking the root of a fraction. We can also write that as taking the root of both the numerator and the denominator. So in this case, it's the fifth rut. So we can take the fifth rut Of Our Numerator 625 P race to the 24th power. And then that is all divided by The fifth ru of our denominator, which is Q raised to the 25th power. Now, we can focus, we'll start with the numerator and simplify that. So when we are simplifying rational expressions, we take a look at the index of our radical, the index here is five that tells us that there needs to be five of any factor in order for it to come out of the radical. So we can break down all of the things inside of our radicals and see if there's five of any factor that get to come out 625. Normally we'd start building a factor tree to see if there's five of any factor inside of it. But we know that 625 is five raised to the fourth power, It needs to be raised to the 5th power if it wants to come out of the radical. So essentially, we can just keep this as the fifth rut Of 625. But that P raise to the 24th power, we can do some work with that. We can get some exponents to the fifth power here, P raised to the 24th could be rewritten as and there are several ways to do this. But the way I'm going to show we have P raise to the fourth power multiplied by pray to the fourth power multiplied by pray to the fourth power multiplied by pray to the fourth power multiplied by pray to the fourth power multiplied by P raised to the fourth power. Or in other words, we have six, these and what we're looking for again is five of a factor. Well, we've got five, we've got five plus one. If we take these first five P raised to the fourth powers, that would be P raised to the fourth power raised to the fifth power. And then just not forgetting that that is still multiplied by the last P raised to the fourth power. So we can write that underneath our radical. We have in the numerator, the fifth rut of 625. And that's multiplied by P raised to the fourth power raised to the fifth power. And then that is multiplied by P raise to the 4th power. That's all being divided by the 5th ro of Q raised to the 25th power. Again, we can expand Q raised to the 25th power. We can rewrite that as Q raised to the fifth power multiplied by Q raise to the fifth power multiplied by Q raise to the fifth power multiplied by Q raise to the fifth power multiplied by Q raised to the fifth power. In this case, we have five QS raised to the fifth power. So this is equal to raise to the fifth power, raise to the fifth power. So in our denominator, we have the fifth rut of Q raised to the 5th power raised to the 5th power. And now anything that has that five and the opponent gets to come outside of the radical. The only things that have something raised to the fifth are P to the fourth in the numerator and Q to the fifth in the denominator. When it comes out, it loses that to the fifth exponent. So in our numerator, we have P raised to the fourth power multiplied by the fifth of 625 P raised to the 4th power. That's what was left inside the radical couldn't come out. And that's all divided by, well, the Q raise to the fifth power gets to come out. So it's just Q raise to the fifth power in the denominator. And this is as simplified as this one is going to get, we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 29–44, simplify using the quotient rule. _______ ⁵√64x¹⁴/y¹⁵

Key Concepts

Quotient Rule

Radical Expressions

Exponent Rules

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