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

Question

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

Accepted Answer

Hello, today we are going to be rationalizing the denominator of the given expression. The expression that is given to us is Z divided by the square root of Y plus seven Z minus two. So what does it mean to rationalize a denominator? Well, in order to rationalize the denominator, we want to simplify the denominator into a form that doesn't contain any square roots. Or in other words, we want to rewrite the denominator without the square root symbols. So how can we start this process? Well, whenever you want to simplify the square root symbols in a denominator, you can multiply the denominator by its conjugate, the conjugate is essentially the opposite of the current denominator. So if we have the square root of Y plus seven Z minus two in the denominator, the conjugate to this quantity will be the square root of Y plus seven Z. And instead of minus plus two, this will be the conjugate opposite to the denominator. If we multiply the denominator by this quantity, we will be able to eliminate the square root symbol. 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 we will multiply the numerator by Y plus seven Z plus two. What we need to do now is we need to simplify the numerator and denominator. Let's go ahead and start with the numerator. In the numerator. We have the quantity Z multiplied by the square root of Y plus seven Z plus two. In order to simplify this numerator, we are going to take the variable Z and distribute it into the conjugate. This will give us Z multiplied by the square root of Y plus seven Z plus two Z. 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, the square root of Y plus seven Z minus two multiplied by the conjugate, which is the square root of Y plus seven Z plus two. In order to simplify the denominator, we are going to foil these quantities together. So we are going to take the first term which is the square root of Y plus seven Z and multiply to the first term of the second group, the square root of Y plus seven Z multiplied by the square root of Y plus seven Z will leave us with the square root of the quantity Y plus seven Z squared. Next, we will take Y plus seven Z and multiply to positive two. This will give us positive two multiplied by the square root of Y plus seven Z. Now we will take negative two in the first group and multiply to the first term in the second group, negative two multiplied by the square root of Y plus seven Z will give us negative two square root of Y plus seven Z. And finally, we will take a negative two and multiply it to positive two which will leave us with negative four. Now, what we need to do is simplify, notice that the middle two terms are opposites of each other. If we take two square root of Y plus seven Z and subtract two square root of Y plus seven Z, these terms will cancel each other out. The first term square root of Y plus seven Z squared will simplify to be Y plus seven Z whoops Y plus seven Z. And now we just add the constant at the end which is negative four. This is going to be the simplest form of the denominator. So now that we have simplified the numerator and denominator, let's take both quantities and plug it back to the original equ uh expression. So we have the numerator Z multiplied by the square root of Y plus seven Z plus two Z divided by the simplified denominator which is Y plus seven Z minus four. There is no further way to simplify this expression. So this is going to be the simplest form of the expression. With a rationalized denominator. And with that being said, the solution to this problem is going to be b 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 each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. a / (√a + b) - 1

Key Concepts

Rationalizing the Denominator

Properties of Square Roots

Algebraic Manipulation

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