rationalize the numerator: √x+h -√x /h?

Question

x+h is uder the radical - square root of x divided by h

Answer 1

I'm betting that you mean this:

(√(x + h) - √x)/h

Both parts are over h, right? not just the √x?

If so, then you can multiply top and bottom by (√(x + h) + √x):

(√(x + h) - √x)/h × (√(x + h) + √x)/(√(x + h) + √x) =

[(√(x + h) - √x)(√(x + h) + √x)]/[h(√(x + h) + √x)]

Then, using FOIL, you have:

[(√(x + h))² - (√x))²]/[h(√(x + h) + √x)]

Which just becomes:

[x + h - x]/[h(√(x + h) + √x)]

And then:

h/[h(√(x + h) + √x)]

And canceling the h's:

1/[(√(x + h) + √x)]

And that's the final answer.

(But, as an extra Calculus note, if you then look at the limit as h approaches 0, you get 1/(2√x)...and then, looking at what you started with, you've just proved that the derivative of √x is 1/(2√x)... If you don't understand that, don't worry about it.)