Find the value of ${\left(6{\left(8^{\frac{1}{3}}+64^{\frac{1}{3}}\right)}^3\right)}^{\frac{1}{4}}$.

Let's simplify the inner terms first:
$$8^{\frac{1}{3}}=(2^3{)}^{\frac{1}{3}}=2$$
$$64^{\frac{1}{3}}=(4^3{)}^{\frac{1}{3}}=4$$
Now, substitute these values back into the expression:
$${\left(6(2+4{)}^3\right)}^{\frac{1}{4}}$$
$${\left(6(6{)}^3\right)}^{\frac{1}{4}}$$
$${\left(6^4\right)}^{\frac{1}{4}}=6$$