Question

Simplify cube root of 5 over fourth root of 5

Answer 1

\(\large {
\cfrac{\sqrt[3]{5}}{\sqrt[4]{5}}
\\ \quad \\
a^{\frac{{\color{blue} n}}{{\color{red} m}}} \implies \sqrt[{\color{red} m}]{a^{\color{blue} n}} \qquad \qquad

\sqrt[{\color{red} m}]{a^{\color{blue} n}}\implies a^{\frac{{\color{blue} n}}{{\color{red} m}}}\qquad and
\\ \quad \\
a^{-{\color{red} n}} \implies \cfrac{1}{a^{\color{red} n}}\qquad \qquad

\cfrac{1}{a^{\color{red} n}}\implies a^{-{\color{red} n}}
\\ \quad \\
thus
\\ \quad \\
\cfrac{\sqrt[{\color{red}{ 3}}]{5^{\color{blue}{ 1}}}}{\sqrt[{\color{red}{ 4}}]{5^{\color{blue}{ 1}}}}\implies \cfrac{5^{\frac{{\color{blue}{ 1}}}{{\color{red}{ 3}}}}}{5^{\frac{{\color{blue}{ 1}}}{{\color{red}{ 4}}}}}\implies \cfrac{5^{\frac{1}{3}}}{1}\cdot \cfrac{1}{5^{\frac{1}{4}}}\implies 5^{\frac{1}{3}}\cdot 5^{-\frac{1}{4}}\implies ?
}\)