To rationalize the denominator of the expression $\frac{20}{\sqrt[3]{25}}$, we aim to eliminate the cube root from the denominator. Here's the step-by-step process:

Step 1: Identify the cube root

The expression is $\frac{20}{\sqrt[3]{25}}$, and we want to eliminate the cube root from the denominator. The cube root of 25 is $\sqrt[3]{25}$, which we can express as:

$\frac{20}{\sqrt[3]{25}} = \frac{20}{25^{1/3}}$

Step 2: Multiply by the necessary factor

To rationalize the denominator, we need to multiply both the numerator and denominator by a factor that will result in a perfect cube in the denominator. Since we already have a cube root, we should multiply the denominator by $\sqrt[3]{25^2}$ (which is $25^{2/3}$) because:

[ \sqrt[3]{25} \times \sqrt[3]{25^2} = \sqrt[3]{25^3} = 25 ]

Step 3: Perform the multiplication

Multiply both the numerator and denominator by $\sqrt[3]{25^2}$:

[ \frac{20}{\sqrt[3]{25}} \times \frac{\sqrt[3]{25^2}}{\sqrt[3]{25^2}} = \frac{20 \times \sqrt[3]{25^2}}{\sqrt[3]{25} \times \sqrt[3]{25^2}} = \frac{20 \times \sqrt[3]{25^2}}{25} ]

Step 4: Simplify

Now, the denominator is simplified to 25. The numerator becomes $20 \times \sqrt[3]{25^2}$. We can leave the answer as:

$\frac{20 \times \sqrt[3]{25^2}}{25}$

This is the rationalized form of the expression.

Would you like a further breakdown of this process or have other questions?

Related questions:

1. What is the value of $\sqrt[3]{25^2}$?
2. How can we rationalize the denominator of a fraction with a square root instead of a cube root?
3. How do you simplify expressions involving cube roots?
4. What is the difference between rationalizing a square root and a cube root?
5. Can this approach be applied to other roots, like the fourth or fifth root?

Tip:

To rationalize the denominator, you always multiply by the same root power that will complete the root in the denominator to become a whole number.