Multiply Fourth and Seventh Roots: Solving ⁴√8 · ⁷√8

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:

$\sqrt[\textcolor{red}{4}]{8}\cdot\sqrt[\textcolor{blue}{7}]{8}= \\ \downarrow\\ 8^{\frac{1}{\textcolor{red}{4}}}\cdot8^{\frac{1}{\textcolor{blue}{7}}}=$

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

$8^{\frac{1}{4}}\cdot8^{\frac{1}{7}}= \\ \boxed{8^{\frac{1}{4}+\frac{1}{7}}}$

Therefore, the correct answer is answer D.