In Exercises 29–44, simplify using the quotient rule.
______
⁴√13...

Question

In Exercises 29–44, simplify using the quotient rule.
______
⁴√13y⁷/x¹²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to use the quotient rule for simplification of the fourth rut of the fraction in the numerator, we have 37 R raised to the seventh power and that's divided by s raised to the 12th power. Our answer choices are answer choice A Are multiplied by the 4th rut of 37 r cubed All divided by S. raised to the 4th power. Answer choice B R multiplied by the fourth rot of 37 R squared all divided by S cubed. Answer trace C R multiplied by the 4th root of R. cubed All divided by 37 s cubed. And answer choice D Are multiplied by the fourth rate of 37 r cubed all divided by S cubed. All right. The quotient rule tells us that when we are taking the root of a fraction, we can rewrite that as taking that same root. So in this case, the fourth threat of both the numerator and the denominator. So in our numerator, we're taking the fourth rate of 37 R raised to the seventh power and that's divided by, in the denominator, we're taking the fourth root of s raised to the 12th power. Now to simplify radicals, we take a look at the index of the radical. In this case, it's four, the four tells us that there needs to be four of any factor in order for it to come outside of the radical. So looking at the inside of our radical in the numerator 37, does that have four of any factor so that it can come out 37 is just one multiplied by 37. It does not have four of any factor. So that remains inside of the fourth rit in our numerator. However, we can do some work with this are raised to the 7th power. We recall from previous lessons that are raised to the seventh power is equal to are raised to the fourth power multiplied by R cubed because when we have like bases, those are the RS being multiplied, we add their exponents together and four plus three equals seven. So we can rewrite this, noting that this R raise to the fourth power does have that fourth power that we were looking for. So underneath our fourth rit in the numerator, we have 37 multiplied by R ray to the fourth power multiplied by R cubed. Can't forget the R cubed And that's all divided by. Well, we have the fourth of s raised to the 12th power, but we can do some work with S raised to the 12th power as well. We want to look for things that have the fourth power and we can rewrite this as S cubed multiplied by S cubed multiplied by S cubed multiplied by S cubed. And that's not to the fourth power. However, there are four of them. So we can say that this is S cubed raised to the fourth power. So in our denominator, we're taking the 4th rut of S Q Raised to the 4th power. And now anything that's raised to the fourth power because of our four index gets to come out of the radical in the numerator. The only thing that gets to come out, we've got this R raised to the fourth power. So that's going to come out as an R because when they come out of the radical, they lose their fourth power. So we have R multiplied by and nothing else could come out. So it's R multiplied by the fourth root of are cubed. Now that's divided by, in the denominator what's raised to the fourth power inside of the fourth rut that's S cubed. So S cubed gets to come out and there's nothing left inside. Everything came out. This is as simplified as this one 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 29–44, simplify using the quotient rule. ______ ⁴√13y⁷/x¹²

Key Concepts

Quotient Rule

Radical Expressions

Exponents and Their Properties

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