Proving the Equality of Two Fractions Using Algebra

[Kara386]

Homework Statement

This is something I need to show in order to solve the question I've been asked. I need to show that
##\sqrt{\frac{N_A}{N_D} } + \sqrt{\frac{N_D}{N_A} } = \frac{N_A + N_B}{\sqrt{N_A N_B}}##
I know these two sides are equal because wolfram alpha says they are, and also because it works if I sub that into my proof. But I'm doing something really really really stupid I think, because I can't get there.

Homework Equations

The Attempt at a Solution

I thought it might be easiest to square this and simplify, then square root. So
##(\sqrt{\frac{N_A}{N_D} } + \sqrt{\frac{N_D}{N_A} })^2 = \frac{N_A}{N_D} + 2\sqrt{\frac{N_A N_B}{N_A N_B}} + \frac{N_D}{N_A}##
I suspect that's the step that's wrong but I don't know why. Carrying on anyway:
##= \frac{N_A}{N_D} + \frac{N_D}{N_A} + 2##
Finding a common denominator:

##= \frac{N_A N_D + N_D N_A + 2N_A N_D}{N_A N_D}##
Then finally square rooting again:
##=2##
So that's very different to what I'm after and there is some really awful algebra mistake in there. Unfortunately I can't find it, any help would be very much appreciated! :)

 

[Kara386]

Oh yes, that is me being pretty stupid. Could just multiply first term by ##\frac{N_A}{N_A}## and the second term by ##\frac{N_D}{N_D}##, then:
##\sqrt{\frac{N_A^2}{N_A N_D}} + \sqrt{\frac{N_D^2}{N_A N_D} }= \frac{N_A}{\sqrt{N_A N_D}} + \frac{N_D}{\sqrt{N_A N_D}}##

 

[BvU]

Finding a common denominator

leads to ##N_A^2 + 2N_A N_D + N_D^2 \over N_A N_D##

 

[Kara386]

leads to ##N_A^2 + 2N_A N_D + N_D^2 \over N_A N_D##

Ah, that too! Oops! Thank you, I was never going to get that.