[tex]\bf ~\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} \\\\\\ ~\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}} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]\bf \cfrac{25\sqrt{5}}{125^{-2}}~~ \begin{cases} 25=&5\cdot 5\\ &5^2\\ 125=&5\cdot 5\cdot 5\\ &5^3 \end{cases}\implies \cfrac{5^2\sqrt[2]{5^1}}{(5^3)^{-2}}\implies \cfrac{5^2\cdot 5^{\frac{1}{2}}}{5^{3(-2)}}\implies \cfrac{5^{2+\frac{1}{2}}}{5^{-6}} \\\\\\ \cfrac{5^{\frac{5}{2}}}{5^{-6}}\implies 5^{\frac{5}{2}}\cdot 5^6\implies 5^{\frac{5}{2}+6}\implies 5^{\frac{17}{2}}[/tex]
