How do you simplify 3^(1/4) * 4^(1/4)?

The answer is $\sqrt[4]{12}$.
Because of the commutative and associative properties of multiplication, you can rewrite ${a}^{x} \cdot {b}^{x}$ as ${\left(a b\right)}^{x}$
Therefore, ${3}^{\frac{1}{4}} \cdot {4}^{\frac{1}{4}} = {12}^{\frac{1}{4}}$.
We also know that ${a}^{\frac{1}{x}} = \sqrt[x]{a}$.
Thus, ${12}^{\frac{1}{4}} = \sqrt[4]{12} = \sqrt[4]{2 \cdot 2 \cdot 3}$.