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The image you uploaded asks to "Rationalize the numerator" of the following expression:

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

Let me walk through the steps to rationalize the numerator.

### Step-by-step solution:
1. The numerator involves a cube root of a fraction: \( \sqrt[3]{\frac{x^2y}{\sqrt{x}}} \).
   
2. First, simplify the expression inside the cube root. Since \( \sqrt{x} = x^{1/2} \), the fraction becomes:
   \[
   \frac{x^2y}{x^{1/2}} = x^{2 - 1/2}y = x^{3/2}y
   \]

3. Now, the numerator is \( \sqrt[3]{x^{3/2}y} \). This can be expressed as:
   \[
   \left( x^{3/2}y \right)^{1/3} = x^{(3/2)(1/3)}y^{1/3} = x^{1/2}y^{1/3}
   \]

   So the numerator simplifies to \( x^{1/2}y^{1/3} = \sqrt{x} y^{1/3} \).

4. Now the expression becomes:
   \[
   \frac{\sqrt{x} y^{1/3}}{5x}
   \]

5. To rationalize the numerator, we need to eliminate the square root \( \sqrt{x} \) from the numerator. Multiply both the numerator and the denominator by \( \sqrt{x} \):
   \[
   \frac{\sqrt{x} y^{1/3}}{5x} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{x y^{1/3}}{5x \sqrt{x}}
   \]

6. Finally, simplify the expression:
   \[
   \frac{y^{1/3}}{5\sqrt{x}}
   \]

Thus, the rationalized form of the expression is:

\[
\frac{y^{1/3}}{5\sqrt{x}}
\]

Let me know if you'd like more details or have any questions.

### Related questions:
1. How do you rationalize a denominator with a cube root?
2. Can this method be applied to higher roots, like the 4th root or 5th root?
3. What other techniques can simplify expressions involving roots?
4. How would you simplify \( \sqrt[4]{x^3y^2} \)?
5. How do you rationalize expressions with mixed radicals, such as \( \sqrt[3]{x} + \sqrt{x} \)?

**Tip:** When rationalizing, always focus on eliminating radicals from either the numerator or the denominator by multiplying by the appropriate conjugate or radical power.

Rationalize the numerator: \(\frac{\sqrt[3]{\frac{x^2y}{\sqrt{x}}}}{5x}\)

Learn how to rationalize the numerator of the expression \(\frac{\sqrt[3]{\frac{x^2y}{\sqrt{x}}}}{5x}\) by simplifying radicals and applying properties of exponents. This step-by-step guide walks through simplification and rationalization techniques for students in Grades 9-12.

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