Question:
If $$N = \frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}$$Find N (This is a subset of a larger question)
My approach:
After rationalizing the denominator, by multiplying fraction with $\frac{\sqrt{\sqrt{5}+1}}{\sqrt{\sqrt{5}+1}}$, I got:
$$\frac{(\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2})*\sqrt{\sqrt{5}+1}}{\sqrt{5}+1}$$
Which is leading me nowhere. The textbook has simply asserted that $$N^2 = 2$$ (Substituting the large fraction with $N$ to save space)
I squared the numerator and got: $$2\sqrt{5}*2*(\sqrt{\sqrt{5}+2}*\sqrt{\sqrt{5}-2})$$
My Question:
How would I further go with this to solve for $N$? or what would be the correct/easier way to find $N$ ?
How would I know when to rationalize denominators, and when to square the fractions, to get the answer ?
Thank you!