To rationalize the cube root of $\frac{4x}{6}$, the goal is to remove the cube root from the denominator (if it’s in the denominator). Since the cube root involves powers of 3, we need to manipulate the expression so that the denominator becomes a perfect cube.

Step-by-step process:

The expression is:

$\sqrt[3]{\frac{4x}{6}}$

1. Simplify the fraction: Simplify $\frac{4x}{6}$ by dividing the numerator and denominator by 2:

   $\frac{4x}{6} = \frac{2x}{3}$

   So the expression becomes:

   $\sqrt[3]{\frac{2x}{3}}$

2. Rationalize the cube root: To rationalize the denominator, we need to make the denominator a perfect cube. The denominator is 3, and the cube root of 3 isn’t a rational number. We need to multiply both the numerator and denominator by $\sqrt[3]{9}$ (since $9 \times 3 = 27$, and $\sqrt[3]{27} = 3$).

   So, we multiply both the numerator and denominator by $\sqrt[3]{9}$:

   [ \frac{\sqrt[3]{2x}}{\sqrt[3]{3}} \times \frac{\sqrt[3]{9}}{\sqrt[3]{9}} = \frac{\sqrt[3]{2x} \cdot \sqrt[3]{9}}{\sqrt[3]{3} \cdot \sqrt[3]{9}} = \frac{\sqrt[3]{18x}}{\sqrt[3]{27}} = \frac{\sqrt[3]{18x}}{3} ]

Final result:

$\frac{\sqrt[3]{18x}}{3}$

This is the rationalized form of $\sqrt[3]{\frac{4x}{6}}$.

Would you like to dive deeper into any part of this, or have more questions? Here are some related questions:

1. How do you rationalize a square root expression?
2. What are the general steps for rationalizing the denominator of a fraction with square or cube roots?
3. How do you simplify cube roots in expressions with variables?
4. What is the significance of rationalizing denominators in real-world applications?
5. How do you handle cube roots of negative numbers?

Tip: When rationalizing cube roots, always look for factors that can turn the denominator into a perfect cube!