[tex] \bf ~~~~~~~~~~~~\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}}
\\\\
-------------------------------\\\\
8^{-2}\cdot \cfrac{3^0-8^{-3}}{4^{-5}}\implies 8^{-2}\cdot \cfrac{1-\frac{1}{8^3}}{\frac{1}{4^5}}\implies 8^{-2}\cdot \cfrac{1-\frac{1}{(2^3)^3}}{\frac{1}{(2^2)^5}} [/tex]
[tex] \bf 8^{-2}\cdot \cfrac{\quad \frac{512-1}{2^9}\quad }{\frac{1}{2^{10}}}\implies (2^3)^{-2}\cdot \cfrac{\quad \frac{511}{2^9}\quad }{\frac{1}{2^{10}}}\implies 2^{-6}\cdot \cfrac{511}{2^9}\cdot \cfrac{2^{10}}{1}
\\\\\\
2^{-6}\cdot \cfrac{511\cdot 2^{10}}{2^9}\implies 2^{-6}\cdot 511\cdot 2^{10}\cdot 2^{-9}\implies 2^{-6}\cdot 511\cdot 2^{10-9}
\\\\\\
2^{-6}\cdot 511\cdot 2\implies 511\cdot 2^{-6}\cdot 2\implies 511\cdot 2^{-6+1}\implies 511\cdot 2^{-5}
\\\\\\
511\cdot \cfrac{1}{2^5}\implies \cfrac{511}{32} [/tex]
