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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.

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Learn how to simplify the cube root expression \(\sqrt[3]{a^5 \cdot b^4}\) by breaking it down step by step. This algebraic problem involves radicals, exponentiation, and simplification techniques. Suitable for students in Grades 10-12.

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