In Exercises 75–92, rationalize each denominator. Simplify, if po...

Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible.

2√6 + √5
--------------
3√6 - √5

Accepted Answer

Hello, today we're gonna be rationalizing the denominator of the given expression. So what we are given is four times the square root of seven plus the square root of two, all over five times the square root of seven minus the square root of two. Now keep in mind that we are asked to rationalize the denominator. And what that means is that we want to rewrite the denominator without the square roots. In order to do this, we're gonna multiply both the numerator and denominator by the opposite of the denominator. So in the denominator, we have the five times the square root of seven minus the square root of two. And the opposite of this quantity is going to be five times the square root of seven plus the square root of two. And we're gonna multiply both the numerator and denominator by this quantity. So let's go ahead and simplify the numerator and denominator individually. So in the numerator, we have the quantity four times the square root of seven plus the square root of two, multiplied by five times the square root of seven plus the square root of two. And what we're going to do is we're going to foil these two quantities together. So we're gonna take four times the square root of seven and distribute it to each term in the second group. So four times the square root of seven times five times the squared of seven is going to give us 20 times the square root of seven squared and four times the square root of seven times the square root of two is going to give us four times the square root of seven times two. And we're gonna repeat this process for the square root of two and distribute it to each term in the second group. So the square root of two times five times the squared of seven is going to give us five times the square root of seven times two. And the square root of two times the square root of two is going to give us the square root of two squared. Now, what we need to do is we need to simplify the quantities. So the square root of seven squared is going to simplify it to give a seven and 20 times seven is going to give us 140. And the two middle two terms, we have four times the square root of seven times two and five times the square root of seven times two. We can combine both of these quantities into a single quantity by combining the coefficients together. And four plus five is going to give us nine and seven times two is going to give us 14. So this is going to simplify to give us nine times the square root of 14 and the square root of two squared is going to simplify to just give us two. So the numerator is gonna simplify to give us 100 and 40 plus nine times the square root of 14 plus two, which itself is going to simplify to give us 100 and 42 plus nine times the square root of 14. Now let's go ahead and simplify the denominator. In the denominator, we have five times the square root of seven minus the square root of two, multiplied by five times the square root of seven plus the square root of two. So again, we're gonna to foil these two quantities together. So we're gonna take five times the squared of seven and distribute it to each term in the second group. Five times the squared of seven times five times the square of seven is going to give us 25 times the square root of seven squared and five times the squared of seven times the squared of two is going to give us positive five times the square root of 14 and repeat this process for negative square root of two. So negative square root of two times five square root of seven is going to give us negative five times the square root of 14 and negative square root of two times positive square root of two is going to give us negative square root of two squared five square to 14 minus five square to 14 is going to cancel each other out. And again, the squared of seven squared is going to simplify to give us seven and 25 times seven is going to simplify to give us 175 and negative square root of two squared is going to simplify to give us negative two. So what we are left with is 100 and 75 minus two, which is going to give us 173. So we have the simplified form of the numerator and denominator. So let's go ahead and combine it back into the expression in the numerator. We have 142 plus nine times the square of 14 over the denominator which is 173. 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 c 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 75–92, rationalize each denominator. Simplify, if possible. 2√6 + √5 -------------- 3√6 - √5

Key Concepts

Rationalizing the Denominator

Simplifying Radicals

Combining Like Terms

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