# What is sqrt{-sqrt3 + sqrt (3 + 8 sqrt (7 + 4 sqrt3?

Jan 5, 2015

If one may use a calculator, its 2
If no calculator is allowed, then one would have to play around with the laws of surds and use algebraic manipulation to simplify it.

Goes this way:

$\sqrt{7 + 4 \sqrt{3}} = \sqrt{4 + 2 \cdot 2 \sqrt{3} + 3} = \sqrt{{2}^{2} + 2 \cdot 2 \sqrt{3} + {\sqrt{3}}^{2}} = \sqrt{{\left(2 + \sqrt{3}\right)}^{2}} = 2 + \sqrt{3}$  { This is using the identity (a + b)^2 = a^2 + b^2 + 2ab}

$\sqrt{3 + 8 \sqrt{7 + 4 \sqrt{3}}} = \sqrt{3 + 8 \cdot \left(2 + \sqrt{3}\right)} = \sqrt{3 + 16 + 8 \sqrt{3}} = \sqrt{16 + 2 \cdot 4 \sqrt{3} + 3} = \sqrt{{\left(4 + \sqrt{3}\right)}^{2}} = 4 + \sqrt{3}$ { This is using the identity (a + b)^2 = a^2 + b^2 + 2ab}

$\sqrt{- \sqrt{3} + \sqrt{3 + 8 \sqrt{7 + 4 \sqrt{3}}}} = \sqrt{- \sqrt{3} + 4 + \sqrt{3}} = \sqrt{4} = 2$