[tex]\bf a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n}
\qquad \qquad
\sqrt[ m]{a^ n}\implies a^{\frac{ n}{ m}}
\\\\\\
\textit{also recall that }~~~~~~~~~~~~\textit{negative exponents}
\\\\
a^{-n} \implies \cfrac{1}{a^n}
\qquad \qquad
\cfrac{1}{a^n}\implies a^{-n}
\qquad \qquad
a^n\implies \cfrac{1}{a^{-n}}\\\\
-------------------------------[/tex]
[tex]\bf \cfrac{\sqrt{2^{5}}}{\sqrt[3]{2}}\implies \cfrac{\sqrt[2]{2^{5}}}{\sqrt[3]{2^1}}\implies \cfrac{2^{\frac{5}{2}}}{2^{\frac{1}{3}}}\implies 2^{\frac{5}{2}}\cdot 2^{-\frac{1}{3}}\implies 2^{\frac{5}{2}-\frac{1}{3}}
\implies
2^{\frac{15-2}{6}}
\\\\\\
2^{\frac{13}{6}}\implies \sqrt[6]{2^{13}}\implies \sqrt[6]{2^{12+1}}\implies \sqrt[6]{(2^2)^{6+1}}\implies \sqrt[6]{(2^2)^6\cdot 2}
\\\\\\
2^2\sqrt[6]{2}\implies 4\sqrt[6]{2}[/tex]
