Question

\frac{\sqrt{2^{5}}}{\sqrt[3]{2}} Divide these numbers using fractional exponents.

Answer 1

`\bf a^{( n)/( m)} \implies \sqrt[ m]{a^ n} \qquad \qquad \sqrt[ m]{a^ n}\implies a^{( n)/( m)} \\\\\\ \textit{also recall that }~~~~~~~~~~~~\textit{negative exponents} \\\\ a^(-n) \implies \cfrac{1}{a^n} \qquad \qquad \cfrac{1}{a^n}\implies a^(-n) \qquad \qquad a^n\implies \cfrac{1}{a^(-n)}\\\\ -------------------------------`


`\bf \cfrac{\sqrt{2^(5)}}{\sqrt[3]{2}}\implies \cfrac{\sqrt[2]{2^(5)}}{\sqrt[3]{2^1}}\implies \cfrac{2^{(5)/(2)}}{2^{(1)/(3)}}\implies 2^{(5)/(2)}\cdot 2^{-(1)/(3)}\implies 2^{(5)/(2)-(1)/(3)} \implies 2^{(15-2)/(6)} \\\\\\ 2^{(13)/(6)}\implies \sqrt[6]{2^(13)}\implies \sqrt[6]{2^(12+1)}\implies \sqrt[6]{(2^2)^(6+1)}\implies \sqrt[6]{(2^2)^6\cdot 2} \\\\\\ 2^2\sqrt[6]{2}\implies 4\sqrt[6]{2}`