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

Question

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

√a - √b 
√a + √b

Accepted Answer

Hello, today we're gonna be rationalizing the numerator of the given expression. So what we are given is the square root of H plus the square root of K all over the square root of H minus the square root of K. Now, we are told that we want to rationalize the numerator. And what that means is that we want to rewrite the numerator without the square roots. In order to do that, we're going to multiply the numerator and denominator by the opposite of the numerator. So the numerator has the quantity, the square root of H plus the square root of K. The opposite of this quantity is going to be the squared of H minus the square root of K. And we're gonna multiply both the numerator and denominator by the opposite of the numerator. So in the numerator, we have the product of two quantities and we have the product of two quantities in the denominator. Let's go ahead and simplify the numerator and denominator individually. So in the numerator, we have the square root of H plus the square root of K multiplied by the square root of H minus the square root of K. So in order to simplify this quantity, we're going to follow the two quantities together, we're going to take the first term squared of H and distribute it to each term in the second group. So the square root of H times the square root of H is going to give us the square root of H squared and the squared of H times negative squared of K is going to give us negative square root of H K. Then we're going to repeat this process with the second term in the first group and distribute it to each term in the second group. So the square root of K times the square root of H is going to give us positive square root of H K and positive square root of K times negative square root of K is going to give us negative square root of K squared. Now we just need to simplify these terms negative square root of H K plus positive square root of H K is going to cancel each other out. And the squared of H squared is going to simplify to give us H and the squared of K squared is going to simplify to give us K. And what we are left with in the numerator is going to be H minus K. So now let's go ahead and simplify the denominator. In the denominator, we have the quantity the squared of H minus the square root of K multiplied by the square root of H minus the square root of K, just like the numerator, we're going to foil these two quantities together. So we're going to take the first term squared of H and distribute it to each term in the second group. The squared of H times the squared of H is going to give us the square root of H squared and the squared of H times the negative square root of K is going to give us negative square root of H K. Then the negative square, I'm sorry, we're gonna repeat this process with the second term in the first group and distributed to each term in the second group. And the negative squared of K times the squared of H is going to give us negative square root of H K and the negative square root of K times the negative square root of K is going to give us positive square root of K squared. And now we need to simplify. So the squared of H squared is going to simplify to give us H and the squared of K squared is going to simplify to give us K. And we have two instances of the square root of H K. We can combine this into a single term by adding the coefficients together and negative one minus one is going to give us negative two. So both of these terms can combine to give us negative two times the square root of H K. And that means that the denominator is going to simplify to give us H minus two square root of H K plus K. Now that we have the simplified forms of the numerator and denominator, let's go ahead and plug this back into the, into the expression. So the numerator is simplified to H minus K and the denominator is going to simplify to give us H minus two square root of H K plus K. And this is going to be the simplified form of the given expression with its numerator rationalized. And with that being said, the answer to this problem is going to be B 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. √a - √b √a + √b

Key Concepts

Rationalizing the Numerator

Conjugates

Simplifying Radicals

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