Solve: Multiplication of Sixth Root and Cube Root of 12

In order to simplify the given expression we use two laws of exponents:

A. Defining the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

B. The law of exponents for multiplication of terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start from converting the roots to exponents using the law of exponents shown in A:

$\sqrt[\textcolor{red}{6}]{12}\cdot\sqrt[\textcolor{blue}{3}]{12}= \\ \downarrow\\ 12^{\frac{1}{\textcolor{red}{6}}}\cdot12^{\frac{1}{\textcolor{blue}{3}}}=$

We continue, since a multiplication of two terms with identical bases is performed - we use the law of exponents shown in B:

$12^{\frac{1}{6}}\cdot12^{\frac{1}{3}}= \\ \boxed{12^{\frac{1}{6}+\frac{1}{3}}}$

Therefore, the correct answer is answer C.