Question

Find the value of
${(26+15\sqrt{3})}^{\frac{2}{3}}-{(26+15\sqrt{3})}^{-\frac{2}{3}}$

Answer 1

We have $(26+15\sqrt{3})=(8+3\sqrt{3}+12\sqrt{3}+18)=(2+\sqrt{3}{)}^3$

Hence, $(26+15\sqrt{3}{)}^{\frac{2}{3}}-(26+15\sqrt{3}{)}^{-\frac{2}{3}}=(2+\sqrt{3}{)}^2-(2+\sqrt{3}{)}^{-2}$

$=(2+\sqrt{3}{)}^2-(2-\sqrt{3}{)}^2$

$=8\sqrt{3}$