[tex]\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad
\sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}
\\\quad \\% rational negative exponent
a^{-\frac{{ n}}{{ m}}} =
\cfrac{1}{a^{\frac{{ n}}{{ m}}}} \implies \cfrac{1}{\sqrt[{ m}]{a^{ n}}}\qquad\qquad % radical denominator
\cfrac{1}{\sqrt[{ m}]{a^{ n}}}= \cfrac{1}{a^{\frac{{ n}}{{ m}}}}\implies a^{-\frac{{ n}}{{ m}}} \\\\
[/tex]
[tex]\bf -----------------------------\\\\
thus\qquad \sqrt{3}\cdot \sqrt[5]{3}\implies \sqrt[2]{3^1}\cdot \sqrt[5]{3^1}\implies 3^{\frac{1}{2}}\cdot 3^{\frac{1}{5}}
\\\\\\
3^{\frac{1}{2}+\frac{1}{5}}\implies 3^{\frac{7}{10}}\implies \sqrt[10]{3^7}\implies \sqrt[10]{2187}[/tex]
