In Exercises 45–66, divide and, if possible, simplify.
______
³√2...

Question

In Exercises 45–66, divide and, if possible, simplify.
______
³√250x⁵y³
³√2x³

Accepted Answer

Welcome back. I am so glad you're here. We are asked for the following expression, perform division and further simplify if possible. Our numerator is the cub of 108 M cubed N raise to the fifth power that's divided by our denominator, which is the cubed rut of four N ray to the fourth power. Our answer choices are answer choice. A three M cubed rot N answer choice B three N cubed dot M answer choice C three M cubed three N and answer choice D three N cubed three N. All right. Well, looking at our expression, we note that both our numerator and our denominator are being taken the cubed root of. And when we're taking the same root of both the numerator and the denominator, we can rewrite this, taking that root. So in this case, the cubed root of the whole fraction. So we're taking the cubed root of our numerator M cubed N raised to the fifth power divided by our denominator, four N raised to the fourth power. And now we can do the division inside of our radical. So we start off by doing 108 divided by four, 108, divided by four is 27. So now we can set this radical equal to the Cube of the first part of it is 27. Now, for our MS, there's only an M term in the numerator. So we can just carry over our M cubed. And now for our ends, we do have an N in both the numerator and the denominator, we have N to the fifth divided by N to the fourth. We recall from previous lessons that when we are dividing like bases, those are the ends, we are going to subtract their exponents and five minus four equals one. So we're left with a new exponent of one N to the first, which is just the same as N and that's what we have now in our radical, we have the cubed threat of 27 M cubed N and now we look at our index which is three that tells us that for anything to come out of the radical, it needs to have three of its factor. Well, this M there's already three NS there, that one will be able to come out. There's only one N that's not going anywhere. What about 27? Well, we know that 27 is the same as three cubed. So we have three cubed and M cubed and then just this N taking the cubed rut of all of those when we take the cubed root of three cubed, that three gets to come outside of the radical. And it's just three. Now, when we take the cubed rid of M cubed, that M gets to come outside of the radical. It's just an M now and that N isn't going anywhere. So three M is being multiplied by the cubed rot of N. That's 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.

In Exercises 45–66, divide and, if possible, simplify. ______ ³√250x⁵y³ ³√2x³

Key Concepts

Radical Expressions

Simplifying Radicals

Dividing Radicals

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