In Exercises 45–66, divide and, if possible, simplify.
______
√54...

Question

In Exercises 45–66, divide and, if possible, simplify.
______
√54a⁷b¹¹
√3a⁻⁴b⁻²

Accepted Answer

Welcome back. I am so glad you're here. We're asked for the following expression, perform division and further simplify if possible. The expression we're given is in the numerator, we have the square rut of 360 M raised to the fifth power H raised to the ninth power that's divided by, in the denominator, we have the square rut of five M raised to the negative two H raised to the negative six. Our answer choices are answer choice A three M. Cubed, a trace to the 7th power multiplied by The Square Root of two MH. Answer choice B three M ray to the 7th power each cubed multiplied by The Square Root of two MH. Answer choice C six m cubed rays to the 7th power multiplied by The Square Root of two MH. And answer choice D six M rays to the seventh power HQ multiplied by The square root of two MH. All right, we recall from previous lessons that when we are dividing two ruts that are the same rut, we can rewrite this, taking the root of the entire fraction. So in this case, taking the square root of both our numerator which is 360 M raises to the fifth H raised to the ninth divided by our denominator five M raised to the negative two H raises to the negative six. And now we can simplify what's inside of the radical. Before we try to take anything out of the radical, we can start off with 360 divided by five that equals 72. So this is equal to the square root of 72. And now taking M raised to the fifth power divided by M raised to the negative two. We recall from previous lessons that when we are dividing like bases, those are the MS, we are going to subtract their exponents. So in this case, we're taking five minus negative 25 minus negative two is the same as five plus two which equals seven. So the new exponent is seven. This comes out to M raise to the seventh power. We can put that inside of our radical oops. And next we are going to divide our H S. We have H raised to the ninth power divided by H raised to the negative sixth. We have like bases. So we can subtract the exponents. We can take nine minus negative six, which is the same as nine plus six, which equals 15. So our new exponent is 15, we have H raised to the 15th power left. So we've done that division and now we need to further simplify when simplifying a radical. We want to see if what our index is and our index when nothing's written is two that tells us that we need two of any factor in order to pull it outside of the radical. So we need to take a look at all three parts of this to see if they have two of any factor. Let's start with 72, Can be put in a factor tree and that's the same as 36 multiplied by two. And we know 36 is six multiplied by six or in other words, six squared and not forgetting that that's still multiplied by two. But we can rewrite that inside of the radical as we're taking this square re of six squared multiplied by two. now we can look at M raised to the 7th power and see if that has any factors that come in twos. We recall from previous lessons that we could expand M rays to the seventh power and write that as M cubed multiplied by M cubed multiplied by M. Because when we are multiplying like bases, the MS, we're going to add their exponents. So three plus three plus the unwritten one here gives us seven. And I said we were looking for things that are squared in order to come out, but we have two M cubes and M cubed multiplied by M Q is the same as M cubed squared. And then just not forgetting that, that is multiplied by this final M. So we can put that inside of our radical. So we have six squared multiplied by two. And now that's multiplied by AM cute squared and multiplied by M. And now let's take a look at that H raised to the 15th power, so H raised to the 15th power would be the same as each raise to the seventh, multiplied by each raise to the seventh, multiplied by each raise to the first. But we don't have to write the raise to the first. And that gives us two H to the seventh. So this is equal to H to the seventh squared multiplied by H. And now that can go underneath the radical. So six squared multiplied by two multiplied by M cubed squared multiplied by M and multiplied by H raised to the seventh power squared multiplied by H. And now anything that is squared is able to come outside of the radical when we take the square root of it, when it does come out, it loses that squared. So we're taking out a six, an M cubed and an H raise to the seventh power. So we have six M cubed H raise to the seventh power. And that's multiplied by our square root of anything that couldn't come out, which is two mh. And this is as simplified as this one gets. We look at our answer choices and this matches with answer choice. See, well done. We'll catch you on the next one.

In Exercises 45–66, divide and, if possible, simplify. ______ √54a⁷b¹¹ √3a⁻⁴b⁻²

Key Concepts

Radical Expressions

Properties of Exponents

Simplification of Fractions

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