Question

Please show working :)

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) \\\\\\ ~\hspace{7em}\textit{rational exponents} \\\\ a^{( n)/( m)} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-( n)/( m)} \implies \cfrac{1}{a^{( n)/( m)}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] \rule{34em}{0.25pt}`

`\bf \cfrac{25√(5)}{125^(-2)}~~ \begin{cases} 25=&5\cdot 5\\ &5^2\\ 125=&5\cdot 5\cdot 5\\ &5^3 \end{cases}\implies \cfrac{5^2\sqrt[2]{5^1}}{(5^3)^(-2)}\implies \cfrac{5^2\cdot 5^{(1)/(2)}}{5^(3(-2))}\implies \cfrac{5^{2+(1)/(2)}}{5^(-6)} \\\\\\ \cfrac{5^{(5)/(2)}}{5^(-6)}\implies 5^{(5)/(2)}\cdot 5^6\implies 5^{(5)/(2)+6}\implies 5^{(17)/(2)}`