- #1

kalish1

- 99

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It equals $\frac{1}{2},$ and we have tried the following to no avail:

1. Substitution of $x = \sqrt{5}$

2. Substitution of $x = 2\sqrt{5}$

3. Substitution of $x = 5+\sqrt{5}$

4. Substitution of $x = \sqrt{5 + \sqrt{5}}$

Here goes:

$$\dfrac{\dfrac{\sqrt{5 + 2\sqrt{5}}}{2} + \dfrac{\sqrt{5(5 + 2\sqrt{5})}}{4} - \dfrac{\sqrt{10 + 2\sqrt{5}}}{8}}{\dfrac{\sqrt{5(5 + 2\sqrt{5})}}{4} + 5 \cdot \dfrac{\sqrt{5 + 2\sqrt{5}}}{4}}$$

Thanks in advance for any help.

This question has been crossposted here - fractions - How to simplify a diabolical expression involving radicals - Mathematics Stack Exchange