The question asks for the simplified form of the expression:

$\sqrt[3]{a^5 \cdot b^4}$

Let's simplify the expression step by step:

Step 1: Break down the cube root

The cube root can be split across both $a^5$ and $b^4$:

[ \sqrt[3]{a^5 \cdot b^4} = \sqrt[3]{a^5} \cdot \sqrt[3]{b^4} ]

Step 2: Simplify each cube root term

For the term involving $a^5$:

$\sqrt[3]{a^5} = a^{5/3} = a^1 \cdot a^{2/3}$

For the term involving $b^4$:

$\sqrt[3]{b^4} = b^{4/3} = b^1 \cdot b^{1/3}$

Step 3: Combine the simplified terms

Now, combine the results:

$a^1 \cdot b^1 \cdot \sqrt[3]{a^2 \cdot b}$

This simplifies to:

$ab \cdot \sqrt[3]{a^2b}$

Final Answer:

The simplified form is $ab \cdot \sqrt[3]{a^2b}$, which corresponds to option E.

Do you have any questions or would you like further details on this?

Here are 5 related questions to expand your understanding:

1. How do you simplify cube roots of products with variables?
2. What are the general rules for simplifying radicals with exponents?
3. How would the process change if it was a square root instead of a cube root?
4. Can you apply the same method to simplify higher roots, like a fourth or fifth root?
5. How can you simplify expressions with both cube roots and square roots in the same expression?

Tip: Always break down roots into their individual components to simplify complex expressions effectively.