1. ## Rationalizing Numerator

I have a question in which I have to rationalize the numerator but then I end up with radicals in the denominator. It almost seems like a lose-lose:

square root of (x + h) - square root of (x) all divided by h

2. $\displaystyle \frac{\sqrt{x+h} - \sqrt{x}}{h}$

$\displaystyle \bigg(\frac{\sqrt{x+h} - \sqrt{x}}{h}\bigg)\bigg(\frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}\bigg)$

$\displaystyle \frac{x+h -x}{h(\sqrt{x+h} + \sqrt{x})}$

$\displaystyle \frac{1}{\sqrt{x+h} + \sqrt{x}}$

What you doing is taking the derivative of $\displaystyle \sqrt{x}$

3. ^ That's exactly how it's done. You have to have radicals either in the numerator or denominator here, you just can't cancel them. You're right that it is standard convention to not leave any radicals in the denominator but since your question specifically says rationalize the numerator, rules must be bent.

4. So it is okay to have radicals in the denominator. That is what I got I just wasn't sure if I had to rationalize it again or something. Thanks a lot!