To express \( 108 \frac{1}{3} \) in its simplest radical form, you are dealing with the cube root of 108, since \( \frac{1}{3} \) indicates a cube root:

\[
108^{\frac{1}{3}} = \sqrt[3]{108}
\]

### Step-by-Step Simplification:

1. **Prime factorization of 108:**

\[
108 = 2^2 \times 3^3
\]

2. **Simplify the cube root:**

\[
\sqrt[3]{108} = \sqrt[3]{2^2 \times 3^3}
\]

3. You can separate the cube root into:

\[
\sqrt[3]{108} = \sqrt[3]{2^2} \times \sqrt[3]{3^3}
\]

4. Now, simplify further:
   - \( \sqrt[3]{3^3} = 3 \)
   - \( \sqrt[3]{2^2} \) remains as \( \sqrt[3]{4} \).

So, the simplest radical form is:

\[
\sqrt[3]{108} = 3 \times \sqrt[3]{4}
\]

Thus, \( 108^{\frac{1}{3}} = 3 \sqrt[3]{4} \).

Let me know if you'd like further clarification or a detailed explanation!

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### Here are 5 follow-up questions to expand your understanding:
1. How do you calculate cube roots for larger numbers?
2. What is the difference between cube roots and square roots in simplification?
3. Can you express \( 108 \frac{2}{3} \) in simplest radical form?
4. How would you simplify \( 3 \sqrt[3]{32} \)?
5. What role does prime factorization play in simplifying radical expressions?

**Tip:** Always start by breaking the number into its prime factors when simplifying radicals, especially for higher roots like cube roots.

Learn how to express 108^(1/3) in its simplest radical form using cube roots and prime factorization. This step-by-step guide explains how to simplify cube roots, including breaking down 108 as 2^2 * 3^3 and simplifying \( \sqrt[3]{108} \) to 3 * \( \sqrt[3]{4} \). Ideal for students in Grades 9-11.

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