Question

What does “n” mean is equivalent to?

Answer 1

$\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ p^3\cdot q^2\left( \cfrac{p^4\cdot q^n\cdot r^3}{r^{-4}} \right) = p^7q^5r^7 \\\\[-0.35em] ~\dotfill$

$\bf \stackrel{\textit{doing the left-hand-side}~\hfill }{p^3q^2\left( \cfrac{p^4q^n r^3 r^4}{1} \right)\implies} p^3q^2\cdot p^4q^n r^3 r^4\implies p^3p^4q^2 q^n r^3 r^4 \\\\\\ p^{3+4}q^{2+n}r^{3+4}\implies p^7q^{2+n}r^7 \\\\[-0.35em] ~\dotfill\\\\ p^7q^{2+n}r^7 = p^7q^5r^7\qquad \implies \qquad q^{2+n}=q^5\implies 2+n=5\implies n=3$