Cannot for the life of me follow this simplification

http://i.imgur.com/6HvVncv.jpg

I understand how the sqrt comes down, from the right half of the top, but how does the top left part lose the square root?

After simplifying, I can get

sqrt(x^2+1) - x^2 / (x^2+1)^(3/2)

However, the answer that all online calcs give and my tutorial notes give is

1/ (x^2+1)^(3/2)

And I can see where that is going from the working up the top, but can't figure out how the sqrt(x^2+1) turns into (x^2+1)

Re: Cannot for the life of me follow this simplification

One way to see this is to factor, first cancel the 2 with 1/2 at top right then factor

$\displaystyle \frac{(x^2 + 1)^{-\frac{1}{2}}[(x^2 + 1) - x^2]}{x^2 + 1} $

Notice if you distribute, the bases are the same so you add exponents and keep the base , -1/2 + 1 = 1/2 so we're good

Next simplify within brackets = 1 then bring what's left to the denominator so it's exponent is positive, finally keep the same base and add the exponents. Let me know if you need more details.

:)

Re: Cannot for the life of me follow this simplification

Yep,I definitely need more details with that first factoring. I have been sitting, trying to work this out for over two hours

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Re: Cannot for the life of me follow this simplification

Re: Cannot for the life of me follow this simplification

Quote:

Originally Posted by

**lukasaurus** Yep,I definitely need more details with that first factoring. I have been sitting, trying to work this out for over two hours

Let's just look at only the numerator for a moment , after canceling 1/2 with 2 and let's represent the radical with an equivalent exponent.

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

is it easier to see the same base now?

We must factor from both terms because we have subtraction between the 2 terms.

Factoring from the term on the right is easy, just pull it out, factoring from the term on the left is a bit harder but if you can understand it you save time and space. Essentialy , to figure out what exponent to put you subtract the exponents

We identified the common base

$\displaystyle (x^2 + 1) $

We decide to factor out

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

So far we have

$\displaystyle (x^2 + 1)^{-\frac{1}{2}}[ \ \ \ \ \ \ - x^2] $

what do we put in the space? Well, it's going to be the same base with an exponent of...

$\displaystyle \frac{1}{2} - (- \frac{1}{2} ) = 1 $

the positive 1/2 is the exponent on the base that is getting factored the -1/2 is the exponent on the base that is doing the factoring.

$\displaystyle (x^2 + 1)^{-\frac{1}{2}}[(x^2 + 1)^1 - x^2] $

or simply

$\displaystyle (x^2 + 1)^{-\frac{1}{2}}[x^2 + 1 - x^2] $

The rest of it is obvious, i hope to have helped you understand the factoring. Let me know if anything is not clear.

:)