Question

If $\sqrt{37 - 20 \sqrt{3}} + \frac{5 + 2 \sqrt3)} {7+4\sqrt3} = a + b \sqrt3$, find the value of a+b

Answer 1

Step-by-step explanation:

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#