Multiply Fourth Root and Sixth Root of 6: Radical Operations

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 by converting the roots to exponents using the law of exponents shown in A:

$\sqrt[\textcolor{red}{4}]{6}\cdot\sqrt[\textcolor{blue}{6}]{6}= \\ \downarrow\\ 6^{\frac{1}{\textcolor{red}{4}}}\cdot6^{\frac{1}{\textcolor{blue}{6}}}=$

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

$6^{\frac{1}{4}}\cdot6^{\frac{1}{6}}= \\ 6^{\frac{1}{4}+\frac{1}{6}}=$

We continue and perform (separately) the operation of adding the exponents which are in the exponent of the expression in the simplified expression, this is done by expanding each of the exponents to the common denominator - the number 12 (which is the smallest common denominator), then we perform the addition and simplification operations in the exponent's numerator:

$\frac{1}{4}+\frac{1}{6}=\\ \frac{1\cdot3+1\cdot2}{12}=\\ \frac{3+2}{12}=\\ \frac{5}{12}\\$In other words - we get that:

$6^{\frac{1}{4}+\frac{1}{6}}=\\ \boxed{6^{\frac{5}{12}}}$

To summarize the simplification process:

$\sqrt[4]{6}\cdot\sqrt[6]{6}= \\ \downarrow\\ 6^{\frac{1}{4}}\cdot6^{\frac{1}{6}}= \\ 6^{\frac{1}{4}+\frac{1}{6}}=\\ \boxed{6^{\frac{5}{12}}}$

Therefore, the correct answer is answer D.