[tex]~\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]8^x\cdot \cfrac{1}{4^{x-1}}=\cfrac{1}{32}\implies 8^x\cdot \cfrac{1}{4^x4^{-1}}=\cfrac{1}{32}\implies \cfrac{8^x}{4^x4^{-1}}=\cfrac{1}{32} \implies \cfrac{8^x}{4^x}\cdot \cfrac{1}{4^{-1}}=\cfrac{1}{32} \\\\\\ \left( \cfrac{8}{4} \right)^x\cdot 4=\cfrac{1}{32}\implies 2^x\cdot 2^2=\cfrac{1}{32}\implies 2^x\cdot 2^2=\cfrac{1}{2^5}\implies 2^{x+2}=2^{-5} \\\\\\ x+2=-5\implies x=-7[/tex]
