simplify: square root of (a+h) - sqaure root of (a) all divided by h?

Question

(square root of (a+h) - sqaure root of (a)) divided by h

Answer 1

[sqrt(a + h) - sqrt(a)] / h

The first thing you need to do is multiply top and bottom by the conjugate of the numerator. The reason why we want to do this is to get rid of the square root symbols. Recall that the conjugate of
(x - y) is (x + y), and multiplying by a conjugate results in a differences of squares. That is, what we'll end up with is

{ [sqrt(a + h)]^2 - [sqrt(a)]^2 } / [h(sqrt(a + h) + sqrt(a))]

But squaring a square root eliminates the square root symbol, so this becomes

[(a + h) - (a)] / [h(sqrt(a + h) + sqrt(a)]

Note that the numerator has the "a" terms cancel, so we have

h / [h(sqrt(a + h) + sqrt(a)]

And now, the h terms cancel, giving us

1 / [sqrt(a + h) + sqrt(a)]

Answer 2

[sqrt(a+h) - sqrt(a)] / h
I assume you are trying to eliminate the h on the bottom.
Multiply the top and bottom by the conjugate of the numerator.
[sqrt(a+h) - sqrt(a)] [[sqrt(a+h) + sqrt(a)] ]/ [h(sqrt(a+h) +sqrt(a))]
= [(a+h) - a] / [h(sqrt(a+h) +sqrt(a))]
= h / [h(sqrt(a+h) +sqrt(a))]
Cancel the h's
= 1 / (sqrt(a+h) +sqrt(a))