Question

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

Answer 1

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

\(=\sqrt{\dfrac{2+\sqrt{3}}{6}}+\sqrt{\dfrac{2-\sqrt{3}}{6}}\)

\(=\sqrt{\dfrac{4+2\sqrt{3}}{12}}+\sqrt{\dfrac{4-2\sqrt{3}}{12}}\)

\(=\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{12}}+\sqrt{\dfrac{\left(\sqrt{3}-1\right)^2}{12}}\)

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

\(=1\)