To solve the problem, let's extract the relevant information:

Given:

1. Volume of the cylinder (V): $50\pi x^9 y^7 z^3$ cubic centimeters.
2. Height of the cylinder (h): $2xy^3z$ centimeters.
3. Formula for the volume of a cylinder: $V = B \cdot h$, where $B$ is the area of the base.

We need to find the radius (r) of the base of the cylinder. The base area $B$ is the area of a circle, given by $B = \pi r^2$.

Steps:

1. Express the base area $B$: $B = \frac{V}{h}.$

2. Substitute the given values for $V$ and $h$: $B = \frac{50\pi x^9 y^7 z^3}{2xy^3z}.$

3. Simplify the expression for $B$:

   • Coefficient: $\frac{50}{2} = 25$,
   • $x^9 \div x = x^8$,
   • $y^7 \div y^3 = y^4$,
   • $z^3 \div z = z^2$. $B = 25\pi x^8 y^4 z^2.$

4. Relate $B$ to the formula for the base area of a circle $\pi r^2$: $B = \pi r^2 \implies \pi r^2 = 25\pi x^8 y^4 z^2.$

5. Solve for $r^2$: Divide both sides by $\pi$: $r^2 = 25x^8 y^4 z^2.$

6. Solve for $r$: Take the square root of both sides: $r = \sqrt{25x^8 y^4 z^2}.$

   Simplify:

   • $\sqrt{25} = 5$,
   • $\sqrt{x^8} = x^4$,
   • $\sqrt{y^4} = y^2$,
   • $\sqrt{z^2} = z$.

   $r = 5x^4 y^2 z.$

Final Answer:

The radius of the base of the cylinder is: $r = 5x^4 y^2 z \, \text{cm}.$

Would you like a detailed explanation of any step or further clarifications?

Related Questions:

1. How do you calculate the surface area of a cylinder using its radius and height?
2. If the height of the cylinder doubles, how does it affect the volume?
3. What happens to the radius if the volume is halved but the height remains constant?
4. How do the powers of variables affect the proportionality of the radius and height?
5. Can this method be applied to non-cylindrical shapes with different formulas?

Tip:

Always ensure units are consistent throughout the calculation to avoid errors.