1. ## Trouble rationalising

A snippet from a calculus question however i am finding the simplifying of a rationalisation hard.

This $\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}$ needs to be rationalised

now i know i multiply both top and bottom by $\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}$

but I'm having mega difficulty actually doing it and getting an equation that actually looks half decent at the end, if someone could just show me step by step, how to really simplify the equation I would be extremely grateful. Thanks.

2. You could simplify the top line to be...

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

$= \frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x+h}\sqrt{x}}$.

3. But if i do that i can't rationalise it.

4. Originally Posted by Rapid_W
A snippet from a calculus question however i am finding the simplifying of a rationalisation hard.

This $\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}$ needs to be rationalised

now i know i multiply both top and bottom by $\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}$

but I'm having mega difficulty actually doing it and getting an equation that actually looks half decent at the end, if someone could just show me step by step, how to really simplify the equation I would be extremely grateful. Thanks.
first understand that you can move the h right into the deminator to simplify things a bit. So first do as deadstar showed you. Then whenever you want to ratyionalize either a denominator or numerator, simply multiply both top and bottom by the conjugate of the one you want to rationalize. I assume here you want to rationalize the numerator so that you can remaove that nasty h from the problem.

* $\frac{1}{h}*\frac{(\sqrt{x}+\sqrt{x+h})}{(\sqrt{x} +\sqrt{x+h})}$

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

= $\frac{1}{x\sqrt{x+h}+\sqrt{x}(x+h)}$

I'll take it one more step for you

as $h\to{0}$ we get

$\frac{1}{2x\sqrt{x}}$