Question

How do you solve this?

Answer 1

$\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{8^?}{8^3}=\cfrac{1}{8^8}\implies \cfrac{8^?}{8^3}=8^{-8}\implies 8^?=8^3\cdot 8^{-8}\implies 8^?=8^{3-8}\implies 8^?=8^{-5}$

$\bf \begin{array}{llll} \textit{if the bases are the same, for}\\ \textit{the equation to be true, the}\\ \textit{exponents must also be the same} \end{array}~\hspace{5em}?=-5 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{1}{r^3}=r^{-7}\cdot r^?\implies r^{-3}=r^{-7+?}\implies \stackrel{\textit{bases are the same, so are the exponents}}{-3=-7+?\implies 4=?}$