Rationalize each denominator. Assume all variables represent nonn...

Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (√7 - 1) / (2√7 + 4√2)

Accepted Answer

Hello, today, we're going to be simplifying the given expression by rationalizing the denominator. So what we are given is the square root of 11 minus 1/6 times the square root of 11 plus five square root of three. Now, in order to rationalize the denominator, we're going to need to multiply the numerator and denominator by the conjugate of the denominator and the conjugate of the denominator is essentially the opposite. So what we are given the conjugate is going to be six times the square root of 11 -5 square root of three. And we're going to multiply the numerator by the conjugate as well. So what we need to do now is multiply the numerator and denominator with the given congregates. And in order to simplify these quantities, we're going to need to foil the enumerators together and the denominators together. So let's start by simplifying our numerator In the numerator were given the square root of 11 -1 times the quantity six square root of 11 -5 square root of three. Foiling this together square root of 11 times six square at 11 is going to give us six times the square root of 11 squared. Then we're going to multiply the square root of 11 with negative five times the square root of three, which is going to give us negative five times the square root of 11 times three. Next, we're gonna multiply negative one with six square root of 11, which is going to give us negative six square root of 11. And finally, we're going to multiply negative one with negative five square root of three, which is going to give us positive five square root of three. Okay. So the square root of 11 squared is going to simplify to give us 11. And the square root of 11 times three is going to give us the square root of 33. So what we are left with is six times 11, which is going to give us 66-5 times the square root of 33 -6 times the square root of 11-plus 5 times the square root of three. Now, none of these terms can be combined any further. So this is going to be the simplified version of our given numerator. Now, we need to go ahead and simplify the denominator. Now, in the denominator, we have six square root of 11 plus five square root of three times the quantity Six square root of 11 -5 square root of three. So once again, in order to simplify this, we're going to need to foil the quantities together six squared of times six squared of 11 is going to give us 30 times the square root of 11 squared, six squared of 11 times negative five square root of three is going to give us negative 30 square root of 11 times three, five squared of three times six squared of 11 is going to give us positive square root of 11 times three and five squared of three times negative five square root of three is going to give us negative 25 square root of three squared. Now, in order to simplify this, notice that we have negative 30 squared of 11 times three plus positive 30 times the square root of 11 times three. If we add these two terms together, these two terms are going to cancel each other out, The square root of 11 squared is going to simplify to give us 11 and the square of three squared is going to simplify to give us three. So what we are left with is 36 times 11 -25 times three, which is going to give us -75, which is equal to 321. And this is going to be the simplified version of our denominator. Now we put all the terms back in our original expression. What we have is 66 -5 square root of 33 -6 sq root of Plus five square root of three all over 321. And this is going to be the simplified form of our given expression. And with that being said, the answer to this problem is going to be be. 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.

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Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (√7 - 1) / (2√7 + 4√2)

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