simplifying a rational expression

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synapsis

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It's been 10 years since my last math class, and a lot of basic concepts elude me. Here is a problem for college algebra that i am having fits with. I know the answer but i still can't work it out. i have the steps in front of me from the book, but i am stuck at one spot.

2x(x^2 - 2)^1/2 + x^2(x^2 - 2)^-1/2

=2x(x^2 -2)^1/2 + x^2 / (x^2 -2)^1/2 I understand that, you move the negative down and make it a denominator....

=2x(x^2 - 2)^1/2(x^2 - 2)^1/2 / (x^2 - 2)^1/2 + x^2 / (x^2 - 2)^1/2 I understand this, you need to find a LCD which is (x^2 - 2) ^1/2

This is where it loses me... if someone could show me the step- the simple step that i am forgetting to get from the above step in the solution to the below step in the solution... I basically don't understand where the first ^1/2 and the (x^2 - 2)^1/2 in the numerator on the left side go to...

=2x(x^2 - 2) + x^2 / (x^2 - 2)^1/2 What am i missing to get this conclusion????

=2x^3 + x^2 - 4x / (x^2 - 2)^1/2 This is the solution... and i understand the simple distributive math here.

Thank you for any help, i hope everything here is accurate i double checked :)
 
\(\displaystyle \L \mbox{ \frac{2x(x^2 - 2)^{\frac{1}{2}}(x^2 - 2)^{\frac{1}{2}}}{(x^2 - 2)^{\frac{1}{2}}} + \frac{x^2}{(x^2 - 2)^{\frac{1}{2}}}\)

First note that just as \(\displaystyle \mbox{ m^a \times m^b = m^{(a + b)} }\) (for \(\displaystyle \mbox{ m \neq 0}\)),

\(\displaystyle \mbox{ (x^2 - 2)^{\frac{1}{2}}(x^2 - 2)^{\frac{1}{2}} = (x^2 - 2)^{\left(\frac{1}{2} + \frac{1}{2}\right)} = (x^2 - 2)^1 = (x^2 - 2)}\)

So we have

\(\displaystyle \L \mbox{ \frac{2x(x^2 - 2)}{(x - 2)^{\frac{1}{2}}} + \frac{x^2}{(x^2 - 2)^{\frac{1}{2}}}\)

There is a common denominator, so we can add the numerators over that denominator, just as \(\displaystyle \mbox{ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}}\):

\(\displaystyle \L \mbox{ \frac{2x(x^2 - 2)}{(x - 2)^{\frac{1}{2}}} + \frac{x^2}{(x^2 - 2)^{\frac{1}{2}}} = \frac{2x(x^2 - 2) + x^2}{(x^2 - 2)^{\frac{1}{2}}}\)
 
2x(x^2 - 2)^1/2 + x^2(x^2 - 2)^-1/2

I always look for ways to reduce typing (so less wieldy);
let a = x^2 - 2 ; then the equation
= 2xsqrt(a) + x^2 / sqrt(a)
= (2ax + x^2) / sqrt(a)

Now substitute back in and you'll get same as Uncle Unco .
 
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