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The problem asks to "Rationalize the numerator," with the expression given as:

\[
\frac{\sqrt[3]{x^2 y}}{5x}
\]

To rationalize the numerator, we need to eliminate the cube root from the numerator. Here's how we proceed:

### Step-by-step Solution:

1. **Expression:**
   \[
   \frac{\sqrt[3]{x^2 y}}{5x}
   \]
   
2. **Multiply numerator and denominator by an appropriate term** to make the numerator a perfect cube. Since the numerator is \(\sqrt[3]{x^2 y}\), we want to multiply by \(\sqrt[3]{x y^2}\) because this will make the powers of both \(x\) and \(y\) inside the cube root reach full cubes.

   Multiply both the numerator and denominator by \(\sqrt[3]{x y^2}\):
   
   \[
   \frac{\sqrt[3]{x^2 y} \cdot \sqrt[3]{x y^2}}{5x \cdot \sqrt[3]{x y^2}}
   \]

3. **Simplify the numerator:**
   \[
   \sqrt[3]{(x^2 y) \cdot (x y^2)} = \sqrt[3]{x^3 y^3} = xy
   \]
   
   Now the numerator becomes \(xy\).

4. **Simplify the denominator:**
   \[
   5x \cdot \sqrt[3]{x y^2}
   \]

5. **Final Expression:**
   The rationalized expression is:
   \[
   \frac{xy}{5x \cdot \sqrt[3]{x y^2}}
   \]

6. **Cancel out the common terms:**
   The \(x\) in the numerator and denominator cancels out, so we are left with:
   \[
   \frac{y}{5 \cdot \sqrt[3]{x y^2}}
   \]

Thus, the rationalized form of the expression is:

\[
\frac{y}{5 \cdot \sqrt[3]{x y^2}}
\]

Would you like further clarification or additional details?

### 5 Related Questions:
1. What does it mean to "rationalize" in the context of cube roots?
2. Can you rationalize a denominator with a fourth root?
3. How do cube roots behave under multiplication?
4. Is it always necessary to rationalize cube roots in expressions?
5. What are the differences between square root and cube root rationalization?

### Tip:
When rationalizing cube roots, always look for the smallest factor that will turn the root expression into a full cube.

Rationalize the numerator of the expression (cube root of x^2 y) / 5x.

Learn how to rationalize the numerator of the expression (cube root of x^2 y) divided by 5x. This step-by-step solution covers cube root multiplication and radical rationalization, suitable for high school algebra students.

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