Solve Fourth and Sixth Roots: Multiplying √⁴3 × √⁶3

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 between factors with the same bases:

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

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

$\sqrt[\textcolor{red}{4}]{3}\cdot\sqrt[\textcolor{blue}{6}]{3}= \\ \downarrow\\ 3^{\frac{1}{\textcolor{red}{4}}}\cdot3^{\frac{1}{\textcolor{blue}{6}}}=$

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

$3^{\frac{1}{4}}\cdot3^{\frac{1}{6}}= \\ \boxed{3^{\frac{1}{4}+\frac{1}{6}}}$

Therefore, the correct answer is answer D.