[tex]\bf ~\hspace{7em}\textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-\frac{ n}{ m}} \implies \cfrac{1}{a^{\frac{ n}{ m}}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\\\ ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]\bf \cfrac{\sqrt{4}}{\sqrt[3]{4}}\implies \cfrac{\sqrt[2]{4}}{\sqrt[3]{4}}\implies \cfrac{4^{\frac{1}{2}}}{4^{\frac{1}{3}}}\implies 4^{\frac{1}{2}}\cdot 4^{-\frac{1}{3}}\implies 4^{\frac{1}{2}-\frac{1}{3}}\implies 4^{\frac{3-2}{6}} \\\\\\ 4^{\frac{1}{6}}\implies (2^2)^{\frac{1}{6}}\implies 2^{2\cdot \frac{1}{6}}\implies 2\frac{1}{3}\implies \sqrt[3]{2}[/tex]
