Question

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

Answer 1

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+4sqrt(3)) = sqrt(4+2*2sqrt(3)+3) = sqrt(2^2+2*2sqrt(3)+sqrt3^2) = sqrt((2+sqrt3)^2) = 2+sqrt3` `{` This is using the identity`(a + b)^2 = a^2 + b^2 + 2ab}`

`sqrt(3+8sqrt(7+4sqrt3)) = sqrt(3+8*(2+sqrt3)) = sqrt(3+16+8sqrt3) = sqrt(16+2*4sqrt3+3) = sqrt((4+sqrt3)^2) = 4+sqrt3` `{` This is using the identity`(a + b)^2 = a^2 + b^2 + 2ab}`

`sqrt(-sqrt3 + sqrt(3+8sqrt(7+4sqrt3))) = sqrt(-sqrt3+4+sqrt3) = sqrt4 = 2`