Rationalize the denominator.

Question

$\frac{5}{\sqrt3-1}$

Accepted Answer

Hello, today we are going to be rationalizing the denominator of the radical expression. The expression given to us is 13 divided by the square root of 17 minus one. So what does it mean to rationalize a denominator? In order to rationalize a denominator, we want to simplify the denominator in such a way that we remove all the square root symbols from the denominator. Notice that we have a quantity in the denominator that involves a square root symbol. In order to simplify this denominator, we will need to multiply the denominator by the conjugate, which is the equal opposite to the current denominator. So if we have the square root of 17 minus one, the conjugate of this quantity will be the square root of 17 plus one. This will be the equal opposite to the denominator. And if we multiply the denominator by this conjugate, it will remove the square root symbols, we do need to be careful though because if we multiply the denominator by the conjugate, we must also multiply the numerator by the conjugate as well. So what we need to do is we need to simplify the numerator and the denominator. Let's start by simplifying the numerator. In the numerator, we have 13 multiplied by the quantity square root of 17 plus one. In order to simplify this numerator, we will distribute 13 into this quantity square root of 17 plus one. 13 multiplied by the square root of 17 will leave us with 13 square root of 17 and 13 multiplied by one will give us 13. So we are left with th 13 square root of 17 plus 13 in the numerator. Now let's simplify the denominator. In the denominator, we have the square root of 17 minus one multiplied by the quantity square root of 17 plus one. In order to simplify these quantities, we will have to foil the two groups together. We will start by taking the square root of 17 in the first group and multiplying it to each term in the second group. The square root of 17 multiplied by the square root of 17 will leave us with the square root of 17 squared square root of 17 multiplied by positive one will give us positive square root of 17. Now we will take negative one in the first group and multiply it to each term in the second group. Negative one multiplied by the square root of 17 will leave us with negative square root of 17 and negative one multiplied by positive one will leave us with negative one. Now we need to simplify square root of 17 minus square root of 17 will give us zero. So the middle two terms of the denominator will eliminate each other out. And the first term square root of 17 squared will simplify to leave us with 17. That means we can rewrite the denominator to be 17 minus one and 17 minus one will give us 16. Now that we've simplified the numerator and denominator, let's put the two into the original expression. This will allow us to rewrite the expression as 13 multiplied by the square root of 17 plus 13 divided by 16. There is no further way to simplify the expression. So this is going to be the simplest form of the expression with a rationalized denominator. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to rationalize an expression. And I'll go ahead and see you all in the next video.

Rationalize the denominator.
$\frac{5}{\sqrt3-1}$

Key Concepts

Rationalizing the Denominator

Conjugates

Simplifying Radicals

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