Question

What is the value of m in the quotient of powers


`\frac{9 {}^( - 3) }{9 {}^(2) } = \frac{1}{9 {}^(m) }`


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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{9^(-3)}{9^2}=\cfrac{1}{9^m}\implies 9^(-3)\cdot 9^(-2)=1\cdot 9^(-m)\implies 9^(-3-2)=9^(-m)\implies 9^(-5)=9^(-m) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill 5=m~\hfill`