Question

Rationalize the denominator and simplify.

Answer 1

\[\frac{ 2\sqrt{x}-3\sqrt{y} }{ \sqrt{x}+\sqrt{y} }\]

Answer 2

You need a "conjugate" to exploit the property: \((a+b)(a-b) = a^{2} - b^{2}\)

You have \(\sqrt{x}+\sqrt{y}\). What do you suppose its "conjugate" might be?

Answer 3

\[\sqrt{x}-\sqrt{y}\]

Answer 4

That's the conjugate. So now what?

Answer 5

Perfect. Multiply both numerator and denominator by that.

The denominator is easy. \(x - y\)

Answer 6

Do what? Lol

Answer 7

The numerator: \((2\sqrt{x}-3\sqrt{y})(\sqrt{x}-\sqrt{y})\)

Answer 8

\[2x+3y-2\sqrt{xy}-3\sqrt{xy}\]

Answer 9

Would this be it?

\[(2\sqrt{x}+3\sqrt{y})(\sqrt{x}+\sqrt{y})\]

Answer 10

So like TK said, once you get the bottom, multiply the conjugate to the top. That \(-2\sqrt{xy}-3\sqrt{xy} \) can be simplified.

Answer 11

You distributed right the first time :)

Answer 12

Would that equation be right?

Answer 13

Yes, now just simplify it, those two can be combined: \(2x+3y-\color{red}{2\sqrt{xy}-3\sqrt{xy}} \)

Answer 14

think of the \(\sqrt{xy} \) as x's if that helps. they're the same and can be combined:

\(2x+3y-\color{red}{2x-3x} \)

Seem easier now??

Answer 15

-3x+3y ?

Answer 16

Oh, sorry about that, x was a terrible sub since it's already there.. err let's make it k:

\(2x+3y-\color{green}{2k-3k} \)

Noww try it :P

Answer 17

-5k+2x+3y

Answer 18

Good :)

So now when we put our fraction back together after taking it apart we end up with this:

\(\Large \frac{2x+3y-5\sqrt{xy}}{x-y} \)

I doubt we can simplify that easily, so that will be what we are looking for :P

Answer 19

oh, yea the \(k=\sqrt{xy} \)

Answer 20

This is what the format of the answer has to look like

Answer 21

Ugh never mind. It won't load.

Answer 22

Sorry about that :/

Answer 23

So anyways, now what? Lol

Answer 24

Since it can't be simplified that'll be your answerr

Answer 25

What is?

Answer 26

http://prntscr.com/6um44i

Answer 27

I have leave now too, hope that made sense, and good luck :)

Answer 28

Thank you so much! :D