Rationalize the denominator.

Question

$\frac{6}{\sqrt5+\sqrt3}$

Accepted Answer

Hello, today we are going to be rationalizing the denominator of the given radical expression. The expression given to us is 10 divided by the square root of 11 plus the square root of 21. So what does it mean to rationalize a denominator? Well, in order to rationalize the denominator, we want to rewrite the denominator in a form that has no square root symbols in the denominator. So how can we start this process? Well, whenever you want to rationalize a denominator, and in this case, we have a quantity that involves square roots, what we could do is we can multiply by the conjugate of the denominator. The conjugate is going to be the opposite of the current denominator. So if we have the square root of 11 plus the square root of 21 then the conjugate to this quantity will be the square root of 11 minus the square root of 21. If we multiply the conjugate to the denominator, this will eliminate the square roots. But we have to be careful 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 now is we need to simplify the numerator and denominator. Let's go ahead and start by simplifying our numerator. In the numerator, we have the product of 10 and the square root of 11 minus the square root of 21. What we could do is we can take the number 10 and distribute it to each term in this quantity that will give us 10 multiplied by the square root of 11 minus 10, multiplied by the square root of 21. There's no further way to simplify the numerator. So we will leave it as is let's go ahead and now simplify the denominator. In the denominator, we have the quantity square root of 11 plus the square root of 21 multiplied by the quantity square root of 11 minus the square root of 21. In order to simplify the denominator, we're going to need to foil the two quantities together to start, we will take the square root of 11 in the first term and multiply to each term in the second group. So the square root of 11 multiplied by the square root of 11 will give us the square root of 11 squared, the square root of 11 multiplied by negative square root 21 can be written as the negative square root 11 multiplied by 21. We will then take the second term in the first group and multiply to each term in the second group. Square root of 21 multiplied by positive square root of 11 will give us positive square root 11 multiplied by 21. Finally, positive square root 21 multiplied by negative square root. 21 will leave us with negative square root of 21 squared. Now, what we have to do is we have to combine like terms and simplify the negative square root of 11 multiplied by 21 plus positive square root. 11 multiplied by 21 will cancel each other out. The square root of 11 squared can simplify to be 11 and negative square root of 21 squared will simplify to be negative 21. Finally, the difference between 11 and 21 will leave us with negative 10. This will be the simplest form of the denominator. Now that we've simplified the numerator and denominator, let's go ahead and plug these values back into the expression. So the new numerator for the expression will be 10 multiplied by the square root of 11 minus 10, multiplied by the square root of 21. And this will be divided by negative 10. Notice that we have a factor of 10 in the denominator and we could factor out 10 in the numerator. If we factor out 10 in the numerator, we can rewrite this to be 10 multiplied by the square root of 11 minus the square root of 21 divided by negative 10, since we have a multiple of 10 in both the numerator and denominator, we can simplify these to be one that will leave us with negative one multiplied by the square root of 11 minus the square root of 21. Finally, we will distribute the negative one into this quantity. And that will give us negative square root of 11 plus the square root of 21. And if we reorder these terms to have the positive term, first, the simplest form of this expression will be the square root of 21 minus the square root of 11. This is going to be the simplest form of the original 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 of how to rationalize a denominator. And I'll go ahead and see you all in the next video.

Rationalize the denominator.
$\frac{6}{\sqrt5+\sqrt3}$

Key Concepts

Rationalizing the Denominator

Conjugates

Simplifying Radicals

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