Question

$\sqrt{\mathbf{(3 +}\sqrt{\mathbf{5}}\mathbf{)}}$is equal to

• A. $\sqrt{5} + 1$
• B. $\sqrt{3} + \sqrt{2}$
• C. $(\sqrt{5} + 1)/\sqrt{2}$
• D. $\frac{1}{2}(\sqrt{5} + 1)$

Answer 1

Answer: (C)

$(\sqrt{5} + 1)/\sqrt{2}$

Answer 2

Let $\sqrt{3 + \sqrt{5}} = \sqrt{x} + \sqrt{y}$

$3 + \sqrt{5} = x + y + 2\sqrt{xy}$. Obviously $x + y = 3$ and $4xy = 5$.

So $(x - y)^{2} = 9 - 5 = 4$or $(x - y) = 2$

After solving $x = \frac{5}{2},y = \frac{1}{2}$. Hence

$\sqrt{3 + \sqrt{5}} = \sqrt{\frac{5}{2}} + \sqrt{\frac{1}{2}} = \frac{\sqrt{5} + 1}{\sqrt{2}}$