Solve: Multiplication of Sixth Root and Cube Root of 12

Converting radicals to fractional exponents makes it easier to apply exponent laws! $\sqrt[6]{12}$ becomes $12^{\frac{1}{6}}$, which clearly shows we can use the multiplication rule for same bases.

Find a common denominator! Since $\frac{1}{3} = \frac{2}{6}$, we get $\frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$. Always simplify your final fraction!

If the bases are different (like $12^{\frac{1}{6}} \cdot 8^{\frac{1}{3}}$), you cannot add the exponents! The multiplication rule $a^m \cdot a^n = a^{m+n}$ only works when the bases are identical.

Those would be the answers if we were asked for just one of the radicals! But we're multiplying them together, so we need to use the exponent addition rule to get $12^{\frac{1}{6} + \frac{1}{3}}$.

The question asks for the exact form, so $12^{\frac{1}{6} + \frac{1}{3}}$ is the complete answer. You could simplify the exponent to $12^{\frac{1}{2}}$ or even $\sqrt{12}$, but the addition form shows your work clearly!