The given expression is $a^2 \cdot \sqrt[4]{a^3}$, and the task is to bring all terms under a single root.

To achieve this, we can rewrite the expression in a single fourth root:

1. Rewrite $a^2$ in terms of a fourth root: $a^2 = \sqrt[4]{a^8}$.
2. Combine the terms under one root: [ a^2 \cdot \sqrt[4]{a^3} = \sqrt[4]{a^8} \cdot \sqrt[4]{a^3} = \sqrt[4]{a^{8+3}} = \sqrt[4]{a^{11}}. ]

The correct answer is $\sqrt[4]{a^{11}}$, which corresponds to option (a).

Would you like more details or have any questions?

Here are five related questions to further explore this concept:

1. How do you simplify expressions with different root indices?
2. What are the rules for combining exponents under a root?
3. How do you convert expressions from one root to another?
4. Why does $a^2$ convert to $\sqrt[4]{a^8}$?
5. How can you simplify expressions involving products and roots more efficiently?

Tip: When combining expressions with different roots, try converting each term to the same root before merging.