Math Help - Cannot for the life of me follow this simplification

1. Cannot for the life of me follow this simplification

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)

2. 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

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

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.

3. 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

5. Re: Cannot for the life of me follow this simplification

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.

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

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

$(x^2 + 1)$

We decide to factor out

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

So far we have

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

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

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

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.

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

or simply

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

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