In Exercises 93–104, rationalize each numerator. Simplify, if pos...

Question

In Exercises 93–104, rationalize each numerator. Simplify, if possible.

³√2x 
³√y

Accepted Answer

Hello, today we're gonna be rationalizing the numerator of the given expression. So what we are given is the Q root of four U over the Q root of five V. Now keep in mind we want to go ahead and rationalize the numerator. And what that means is that we want to rewrite the numerator without the cube root. But that means we need to multiply the numerator and denominator by a quantity that's going to allow us to eliminate the cube root in the numerator in order to do this, let's make note of two things. First, if we have the cube root of a term cubed, for example, the cube root of a cubed, this is going to simplify to just give us a. In order to approach this, we can use the property of square root multiplication. And that states that if you have the M third of a multiplied by the M through of B, as long as the numbers in front of the square root symbols are the same, you can combine this into a single square root by taking the product of the inside arguments. Or in other words, the M root of A times the M through to B can be simplified as the mth root of A times B. So what we wanna do is we want to multiply the cube root of four U by a quantity that's going to eliminate the cube root. And that quantity in this case is going to be the cube root of four U squared. And we're going to multiply this quantity to both the numerator and denominator. Now, you may be wondering why are we using the Q root of four U squared? Well, let's go ahead and take a look at just the numerator. For now, in the numerator, we have the product, the cube root of four U multiplied by the cube root of four U squared. If we apply the multiplicative property for square roots to the numerator, then we can combine the product of these two cube roots into a single cube root and rewrite this as the cube root of four U times the quantity four U squared. And since we're multiplying four U to four U squared, we could combine this into a single term and rewrite the numerator as the cube root of four U cubed and applying the cube root property, which I've written up here, we can simplify the cube root of four U cubed to just give us the quantity for you. And that's going to allow us to rewrite the numerator without the need of the key root. So now that we know that the cube root of four U squared works. Let's go ahead and simplify the denominator. So in the denominator, we have the Q root of five V multiplied by the Q root of four U squared. If we apply the multiplicative property to the denominator, we can combine this into a single Q root and rewrite this as the Q root of five V multiplied by four U squared. And for you squared can be simplified to give us 16 U squared. So we have the Q root of five V times 16 U squared and five times 16 is going to simplify to give us 80. So we have the cube root of 80 U squared V. Now there's a few more simplifications that we can do to the denominator. See we can rewrite 80 as the product of prime numbers. And we can go ahead and rewrite 80 as the cube root of two times two times, two times 10 times U squared times V. And notice that we have two times itself, three times, we can rewrite this quantity to give us two cubed. And the cube root is going to allow us to remove this two as a single two in front of the cube root. So we can simplify the denominator to give us two times the cube root of 10 U squared V. And this is going to be the simplest form of the denominator. So now that we have the simplest form of the numerator and the simplest form of the denominator, let's go ahead and plug this back into the original expression. So we have four U over two times the Q root of 10 U squared times V. And notice that four and two are both multiples of two. So we can go ahead and reduce this to give us 2/1. And that means that the simplest form of the expression is going to be to you over the cube root of 10 U squared V. And this is going to be the rationalized form of the given expression. And with that being said, the answer to this problem is going to be D so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 93–104, rationalize each numerator. Simplify, if possible. ³√2x ³√y

Key Concepts

Rationalizing the Numerator

Simplifying Radicals

Properties of Exponents and Radicals

Watch next