How does $\sqrt {\frac{{4 + \sqrt {15} }}{8}} = \frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$

Question

I have the follow answering to a question from my textbook: $\sqrt {\frac{{4 + \sqrt {15} }}{8}}$ However my textbook simplifies it to: $\frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$

I've checked and my answer seems to be the same value as the one in the textbook but How do I go about getting the textbooks answer?

Answer 1

Multiply as follows:

$$\sqrt{\frac{4+\sqrt{15}}{8}\frac{2}{2}}=\sqrt{\frac{8+2\sqrt{15}}{16}}=\frac{\sqrt{8+2\sqrt{15}}}{4}$$

Answer 2

$$\frac{4+\sqrt{15}}8=\frac{8+2\sqrt{15}}{16}$$

which in fact can be further simplified as

$$\frac{(\sqrt3)^2+(\sqrt5)^2+2\sqrt3\sqrt5}{16}=\left(\frac{\sqrt5+\sqrt3}4\right)^2$$