Question

\(\sqrt{\dfrac{3\sqrt{3}-4}{2\sqrt{3}+1}}-\sqrt{\dfrac{\sqrt{3}+4}{5-2\sqrt{3}}}\)

Answer 1

\(=\sqrt{\dfrac{\left(3\sqrt{3}-4\right)\left(2\sqrt{3}-1\right)}{11}}-\sqrt{\dfrac{\left(\sqrt{3}+4\right)\left(5+2\sqrt{3}\right)}{13}}\\ =\sqrt{\dfrac{22-11\sqrt{3}}{11}}-\sqrt{\dfrac{13\sqrt{3}+26}{13}}\\ =\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}\\ =\sqrt{\dfrac{4-2\sqrt{3}}{2}}-\sqrt{\dfrac{4+2\sqrt{3}}{2}}\\ =\sqrt{\dfrac{\left(\sqrt{3}-1\right)^2}{2}}-\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{2}}\\ =\dfrac{\sqrt{3}-1-\sqrt{3}-1}{\sqrt{2}}=\dfrac{-2}{\sqrt{2}}=-\sqrt{2}\)