Rationalize the denominator.

Question

$\frac{3}{3+\sqrt7}$

Accepted Answer

Hello, today we are going to be rationalizing the denominator of the given radical expression. The expression given to us is 23 divided by 23 plus the square root of 19. So how can we rationalize the denominator of this radical expression? Well, in order to rationalize, we want to simplify the denominator in a way that allows us to remove the square root symbols from the denominator. So notice that we have a quantity in the denominator involving a square root. In order to remove this square root, we want to multiply the denominator by its conjugate or in other words, it's equal opposite. So if we have 23 plus the square root of 19 in the denominator, then the conjugate of this quantity will be 23 minus the square root of 19. So if we multiply this conjugate to the denominator, this will allow us to remove the square root symbols. 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 now what we need to do is we need to simplify the numerator and denominator. Let's go ahead and start by simplifying the numerator. In the numerator, we have 23 multiplied by the quantity 23 minus the square root of 19. In order to simplify this numerator, we will take 23 and distribute it to the quantity 23 minus 19 square root of 19. So 23 multiplied by 23 will give us the product 529 and 23 multiplied by negative square root of 19 will give us negative 23 multiplied by the square root of 19. This is going to be the simplest form of the numerator. So now let's go ahead and simplify the denominator. In the denominator, we have the quantity 23 plus the square root of 19 multiplied by 23 minus the square root of 19. In order to simplify this denominator, we will have to foil the two quantities together. So what we are going to do is we are going to take the first term of the first group 23 and multiply it to each term in the second group 23 multiplied by 23 will give us the product 529 23 multiplied by negative square root of 19 will give us negative 23 multiplied by the square root of 19. Next, we will take the square root of 19 in the first group and multiply to each term in the second group, square root of 19 multiplied by 23 will give us positive 23 square root of 19 and positive square root of 19 multiplied by negative square root of 19 will leave us with negative square root of 19 squared. Now, what we need to do is simplify, notice that the middle two terms are equal opposites of each other. Negative 23 square root of 19 plus 23 square root of 19 will give us zero. So these terms will cancel each other out. The last term squared of 19 squared will simplify to give us the S square just 19. So we are left with 529 minus 19. And the difference of these two terms will give us the value 510. Now that we have the numerator and denominator simplified, we will plug this back into the original expression. This will allow us to rewrite the expression as 529 minus 23 square root of 19 divided by 510. This is going to be the simplest form of the radical expression with a rationalized denominator. And with that being said, the solution to this problem is going to be D. So I hope this video helps you in understanding how to rationalize a denominator. And I'll go ahead and see you all in the next video.

Rationalize the denominator.
$\frac{3}{3+\sqrt7}$

Key Concepts

Rationalizing the Denominator

Conjugates

Properties of Square Roots

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