Question

how do you solve this?

Answer 1

$\mathsf{Let : P = (\sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}})}$

$\mathsf{Squaring\;on\;Both\;Sides,\;We\;get:}$

$\mathsf{\implies P^2 = 12 + 6\sqrt{3} + 12 - 6\sqrt{3} + 2\sqrt{(12 + 6\sqrt{3})(12 - 6\sqrt{3})}}$

$\mathsf{\implies P^2 = 24 + 2\sqrt{[(12)^2 - (6\sqrt{3})^2]}}$

$\mathsf{\implies P^2 = 24 + 2\sqrt{[(144 - 108]}}\\\\\mathsf{\implies P^2 = 24 + 2\sqrt{36}}$

$\mathsf{\implies P^2 = 24 + 12}$

$\mathsf{\implies P^2 = 36}$

$\mathsf{\implies P = 6}$