<s>[INST] $\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2}{\sqrt{1+\frac{1}{\sqrt{x}}}+\sqrt{1-\frac{1}{\sqrt{x}}}}$? According to an example in my text book:
$$\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2}{\sqrt{1+\frac{1}{\sqrt{x}}}+\sqrt{1-\frac{1}{\sqrt{x}}}}$$
I don't see how this works. The closest I can get is:
$$\frac{(x + \sqrt{x}) - (x-\sqrt{x})}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}} = \frac{2\sqrt{x}}{\sqrt{x}(\sqrt{\frac{1}{\sqrt{x}} + 1}+\sqrt{\frac{1}{\sqrt{x}} -1})} = \frac{2}{\sqrt{\frac{1}{\sqrt{x}} + 1}+\sqrt{\frac{1}{\sqrt{x}} -1}}$$
Which is slightly off. But I'm not even sure if I calculated that correctly. What am I missing, what's the way to think?
 [/INST] $$ \frac{(x + \sqrt{x}) - (x - \sqrt{x} )}{\sqrt{x + \sqrt{x}} + \sqrt{x - \sqrt{x}}} = \frac{2 \sqrt{x}}{\sqrt{x + \sqrt{x}} + \sqrt{x - \sqrt{x}}}$$
Divide by $\sqrt{x}$ throughout. It simplifies to
$$\frac{2}{\dfrac{\sqrt{x + \sqrt{x}} }{\sqrt{x}} + \dfrac{\sqrt{x - \sqrt{x}}}{\sqrt{x}}}.$$
Now the denominator simplifies to
$\sqrt{\dfrac{x + \sqrt{x}}{x}} + \sqrt{\dfrac{x - \sqrt{x}}{x}} $ and the answer comes out nicely.
</s>