Exponents Simplification

Question

I need assistance with simplifying this rational expression. The expression is $$\frac{1}{\sqrt{29 + 12\sqrt{5}}}.$$

The answer, for reference, is $$\frac{2\sqrt{5} - 3}{11}.$$ Thank you!

Answer 1

Hint:

$$ 29 + 12\sqrt{5} = 3^2 + 2^2\cdot 5 + 2\cdot 3\cdot 2\sqrt{5} = (3 + 2\sqrt{5})^2 $$

Thus

$$ \sqrt{29+12\sqrt{5}} = 3 + 2\sqrt{5} $$

If you can't easily see this, here's an algebraic approach: Suppose $\exists a,b \in \mathbb{Z}$ such that

$$ \sqrt{29+12\sqrt{5}} = a + b\sqrt{5} $$

Squaring both sides gives $$ 29 + 12\sqrt{5} = a^2 + 5b^2 + 2ab\sqrt{5} $$

Since $a$ and $b$ are integers, it must be true that $$ a^2 + 5b^2 = 29, \ 2ab = 12 $$

Solving this gives $(a,b) = (3,2), \ (-3,-2)$. We pick the solution pair that satisfies $a + b\sqrt{5} > 0$, which is $(3,2)$