Question

Answer 1

15.

\(\sqrt[5]{5\sqrt[4]{25.\sqrt[3]{5}}}=\sqrt[5]{5\sqrt[4]{5^2.5^{\dfrac{1}{3}}}}=\sqrt[5]{5\sqrt[4]{5^{\dfrac{7}{3}}}}=\sqrt[5]{5.5^{\dfrac{7}{12}}}=\sqrt[5]{5^{\dfrac{19}{12}}}\)

\(=5^{\dfrac{19}{60}}\Rightarrow m=19;n=60\Rightarrow m+n=79\)

16.

\(=\dfrac{2^{2+\sqrt{5}}.3^{2+\sqrt{5}}.2^{1-\sqrt{5}}}{3^{3+\sqrt{5}}}=2^{2+\sqrt{5}+1-\sqrt{5}}.3^{2+\sqrt{5}-\left(3+\sqrt{5}\right)}\)

\(=2^3.3^{-1}=\dfrac{8}{3}\Rightarrow m+n=11\)

17.

\(2^{\dfrac{1}{3}}.2^{-\dfrac{3}{4}}:2^{-\dfrac{1}{6}}=2^{\dfrac{1}{3}-\dfrac{3}{4}+\dfrac{1}{6}}=2^{-\dfrac{1}{4}}\Rightarrow m=-1;n=4\)

\(\Rightarrow mn=-4\)