The given expression is:

$\frac{3}{\sqrt[3]{24}}$

To rationalize the denominator, we need to eliminate the cube root from the denominator. Here's how we can do it:

Steps to rationalize the denominator:

1. The denominator is $\sqrt[3]{24}$. We want to multiply both the numerator and the denominator by the cube root of a number that will make the denominator a whole number. We need to find a factor that will make $\sqrt[3]{24}$ a perfect cube.

2. Since $24 = 2^3 \times 3$, we multiply both the numerator and denominator by $\sqrt[3]{3}$, which is the cube root of 3.

3. Now multiply:

[ \frac{3}{\sqrt[3]{24}} \times \frac{\sqrt[3]{3}}{\sqrt[3]{3}} = \frac{3 \times \sqrt[3]{3}}{\sqrt[3]{24} \times \sqrt[3]{3}} ]

4. The denominator becomes:

[ \sqrt[3]{24 \times 3} = \sqrt[3]{72} ]

5. So, the expression becomes:

[ \frac{3 \times \sqrt[3]{3}}{\sqrt[3]{72}} ]

This is the rationalized form of the original expression.

Would you like further details or have any other questions?

Related questions:

1. How can we simplify cube roots in expressions?
2. What is the general method for rationalizing higher-order roots like square or cube roots?
3. How do you evaluate cube roots of numbers such as 24 or 72?
4. Can you apply the same rationalization process to square roots or other radical expressions?
5. What is the significance of rationalizing denominators in mathematics?

Tip: When rationalizing cube roots or higher roots, always multiply by the factor that will make the base inside the root a perfect cube (or square for square roots).