Question

Simplify square root of 3 multiplied by the fifth root of 3.

Answer 1

$\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad \sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}} \\\quad \\% rational negative exponent a^{-\frac{{ n}}{{ m}}} = \cfrac{1}{a^{\frac{{ n}}{{ m}}}} \implies \cfrac{1}{\sqrt[{ m}]{a^{ n}}}\qquad\qquad % radical denominator \cfrac{1}{\sqrt[{ m}]{a^{ n}}}= \cfrac{1}{a^{\frac{{ n}}{{ m}}}}\implies a^{-\frac{{ n}}{{ m}}}$

$\bf -----------------------------\\\\ thus\qquad \sqrt{3}\cdot \sqrt[5]{3}\implies \sqrt[2]{3^1}\cdot \sqrt[5]{3^1}\implies 3^{\frac{1}{2}}\cdot 3^{\frac{1}{5}} \\\\\\ 3^{\frac{1}{2}+\frac{1}{5}}\implies 3^{\frac{7}{10}}\implies \sqrt[10]{3^7}\implies \sqrt[10]{2187}$