Question

a)     \(A = \sqrt[3]{{5\sqrt {\frac{1}{5}} }};\,\,a = 5\)

b)    \(B = \frac{{4\sqrt[5]{2}}}{{\sqrt[3]{4}}};\,\,a = \sqrt 2 \)

Answer 1

a)     \(A = \sqrt[3]{{5\sqrt {\frac{1}{5}} }} = \sqrt[3]{{a\sqrt {\frac{1}{a}} }} = \sqrt[3]{{a.{a^{\frac{1}{2}}}}} = \sqrt[3]{{{a^{\frac{3}{2}}}}} = {\left( {{a^{\frac{3}{2}}}} \right)^{\frac{1}{3}}} = {a^{\frac{3}{2}.\frac{1}{3}}} = {a^{\frac{1}{2}}} = \sqrt a \)

b)    \(B = \frac{{4\sqrt[5]{2}}}{{\sqrt[3]{4}}} = \frac{{{2^2}{{.2}^{\frac{1}{5}}}}}{{{4^{\frac{1}{3}}}}} = \frac{{{2^{\frac{{11}}{5}}}}}{{{2^{\frac{2}{3}}}}} = {2^{\frac{{23}}{{15}}}}\)

\(a = \sqrt 2  = {2^{\frac{1}{2}}}\)

=> \(B = {a^{\frac{{23}}{{30}}}}\)