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The problem asks to simplify the radical expression:

\[
\frac{ \sqrt[3]{81x^4} }{ \sqrt[3]{3x^7} }
\]

### Step-by-Step Solution:

1. **Simplify the numerator**:
   \[
   \sqrt[3]{81x^4} = \sqrt[3]{81} \cdot \sqrt[3]{x^4}
   \]
   Since \(81 = 3^4\), we have:
   \[
   \sqrt[3]{81} = \sqrt[3]{3^4} = 3 \cdot \sqrt[3]{3}
   \]
   For the variable term \(x^4\):
   \[
   \sqrt[3]{x^4} = x^{4/3}
   \]
   So the numerator simplifies to:
   \[
   3 \cdot \sqrt[3]{3} \cdot x^{4/3}
   \]

2. **Simplify the denominator**:
   \[
   \sqrt[3]{3x^7} = \sqrt[3]{3} \cdot \sqrt[3]{x^7}
   \]
   For the variable term \(x^7\):
   \[
   \sqrt[3]{x^7} = x^{7/3}
   \]
   So the denominator simplifies to:
   \[
   \sqrt[3]{3} \cdot x^{7/3}
   \]

3. **Form the fraction**:
   Now, divide the simplified numerator by the denominator:
   \[
   \frac{3 \cdot \sqrt[3]{3} \cdot x^{4/3}}{\sqrt[3]{3} \cdot x^{7/3}}
   \]

4. **Cancel common terms**:
   - The \(\sqrt[3]{3}\) cancels out.
   - For the powers of \(x\), subtract the exponents:
     \[
     x^{4/3} - x^{7/3} = x^{(4

Simplify the radical expression: \( \frac{ \sqrt[3]{81x^4} }{ \sqrt[3]{3x^7} } \)

Learn how to simplify the radical expression \( \frac{ \sqrt[3]{81x^4} }{ \sqrt[3]{3x^7} } \) using algebraic techniques, including laws of exponents and radical simplifications. Suitable for high school students in Grades 10-12.

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