To solve the formula for the volume of a sphere, $V = \frac{4}{3} \pi r^3$, for $r$, we need to isolate $r$ on one side of the equation.

Step-by-step solution:

1. Start with the equation: $V = \frac{4}{3} \pi r^3$

2. Multiply both sides of the equation by $\frac{3}{4}$ to get rid of the fraction: $\frac{3}{4} V = \pi r^3$

3. Now, divide both sides by $\pi$ to isolate $r^3$: $\frac{3V}{4\pi} = r^3$

4. Finally, take the cube root of both sides to solve for $r$: $r = \sqrt[3]{\frac{3V}{4\pi}}$

Thus, the solution for $r$ is: $r = \sqrt[3]{\frac{3V}{4\pi}}$

Would you like any further clarification on this?

Here are some additional questions you might find interesting:

1. How does the volume of a sphere change if the radius is doubled?
2. What would happen to the radius if the volume of a sphere is halved?
3. How can you calculate the surface area of a sphere from its radius?
4. What is the relationship between the volume of a sphere and its diameter?
5. How do the formulas for the volume and surface area of a sphere compare?

Tip: Always check the units you are using for the radius and volume to ensure consistency, especially in real-world problems!