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

1 Answer
YouDid · Trevor Ryan. · Sidharth
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+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#

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