Need Help with Rationalizing a Denominator

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maroonblazer
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Joined: Thu Aug 12, 2010 11:16 am
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Need Help with Rationalizing a Denominator

Hi all,
I'm struggling with the following problem from my Algebra 2/Trig text:

Rationalize the denominator and write the fraction in simplest form. All variables represent positive numbers:
$\frac{1}{\sqrt{x}+6}-\frac{2}{\sqrt{6}$

Here's as far as I get:
I multiply the first fraction by 1 using the conjugate, $\sqrt{x}-6$, and the second fraction by $\sqrt{6}$ which gives me:
$\frac{\sqrt{x}-6}{x-36}-\frac{2\sqrt{6}}{6}$

Next, so that I could perform the subtraction, I calculated the common denominator of the two fractions: (x-36)(6), or 6x-216.

That gives me:
$\frac{6(\sqrt{x}-6)}{6x-216} - \frac{2\sqrt{6}(x-36)}{6x-216}$

multiplying out the numerators I get:
$\frac{6\sqrt{x}-36-2x\sqrt{6}-72\sqrt{6}}{6x-216}$

And this is where I'm stuck. The answer given by the text is:
$\frac{3\sqrt{x}+(36-x)\sqrt{6}-18}{3x-108}$

It looks like I’m ok up to the point I get to, but I can't make the last jump(s) to the answer. I think part of my problem is that I thought, when you have binomials or polynomials, you can't cancel or reduce. i.e. you can only do that when you are multiplying factors (not sure if I'm explaining myself clearly there).

Ideas?