Solve the Product: Seventh Root of 4 Times Cube Root of 4

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

a. The definition of root as an exponent:

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

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

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

Let's start by converting the roots to exponents using the law mentioned in a':

$\sqrt[\textcolor{red}{7}]{4}\cdot\sqrt[\textcolor{blue}{3}]{4}= \\ \downarrow\\ 4^{\frac{1}{\textcolor{red}{7}}}\cdot4^{\frac{1}{\textcolor{blue}{3}}}=$

We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b':

$4^{\frac{1}{7}}\cdot4^{\frac{1}{3}}= \\ \boxed{4^{\frac{1}{7}+\frac{1}{3}}}$

Therefore, the correct answer is answer b'.