To simplify the expression:

[ \sqrt[4]{a^2} \cdot \sqrt[4]{a^2} ]

1. Apply the Property of Radicals: The product of two fourth roots can be combined as a single fourth root:

   [ \sqrt[4]{a^2} \cdot \sqrt[4]{a^2} = \sqrt[4]{(a^2) \cdot (a^2)} = \sqrt[4]{a^4} ]

2. Simplify the Result: Since $\sqrt[4]{a^4} = a$, the simplified form of the expression is:

   $a$

Therefore, the simplified expression is $a$.

Would you like more details or have any questions?

Here are some related questions to consider:

1. How would the expression change if it were $\sqrt[3]{a^2} \cdot \sqrt[3]{a^2}$?
2. What happens if you multiply $\sqrt[n]{a^m} \cdot \sqrt[n]{a^m}$ in general?
3. How does the simplification process differ for square roots versus