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To simplify the given radical expression \( 4 \cdot \sqrt[3]{320} \), follow these steps:

### Step 1: Prime factorization of 320
The prime factorization of \( 320 \) is:
\[
320 = 2^6 \cdot 5
\]

### Step 2: Apply cube root to the factors
The cube root of \( 320 \) can be expressed as:
\[
\sqrt[3]{320} = \sqrt[3]{2^6 \cdot 5} = \sqrt[3]{2^6} \cdot \sqrt[3]{5}
\]

Since \( 2^6 = (2^3)^2 \), simplify the cube root:
\[
\sqrt[3]{2^6} = 2^2 = 4
\]

Thus:
\[
\sqrt[3]{320} = 4 \cdot \sqrt[3]{5}
\]

### Step 3: Multiply by 4
Now multiply \( 4 \) by \( 4 \cdot \sqrt[3]{5} \):
\[
4 \cdot \sqrt[3]{320} = 4 \cdot (4 \cdot \sqrt[3]{5}) = 16 \cdot \sqrt[3]{5}
\]

### Final Answer:
\[
4 \cdot \sqrt[3]{320} = 16 \cdot \sqrt[3]{5}
\]

Would you like a detailed explanation or have any further questions? Here are some related questions to deepen understanding:

1. How is cube root simplification different from square root simplification?
2. Can \( \sqrt[3]{320} \) be approximated to a decimal form?
3. What are other examples of simplifying cube roots with higher powers?
4. How do prime factorization and exponents help simplify radicals?
5. Can \( \sqrt[3]{320} \) be simplified further if written in a different form?

**Tip**: Always start with the prime factorization of the number when simplifying radicals to identify powers that match the root index.

Simplify each radical: 4 * cube root of 320

Learn how to simplify the radical expression 4 * cube root of 320 using prime factorization and cube root properties. Step-by-step explanation provided, suitable for students in Grades 8-10.

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