**The Relationship Between X, Y, and Z**

Let's analyze the given information step by step to understand the relationship between X, Y, and Z.

1. X varies as a cube of Y:
- This means that X is directly proportional to the cube of Y.
- Mathematically, we can express this relationship as X ∝ Y^3.

2. Y varies as the 5th root of Z:
- This means that Y is directly proportional to the 5th root of Z.
- Mathematically, we can express this relationship as Y ∝ Z^(1/5).

3. X varies as the nth power of Z:
- This means that X is directly proportional to the nth power of Z.
- Mathematically, we can express this relationship as X ∝ Z^n.

**Finding the Value of n**

To find the value of n, we need to determine the relationship between X and Z. Let's combine the given relationships:

- X ∝ Y^3 (from the first relationship)
- Y ∝ Z^(1/5) (from the second relationship)

Substituting the value of Y from the second relationship into the first relationship, we get:

X ∝ (Z^(1/5))^3

Simplifying further:

X ∝ Z^(3/5)

Comparing this with X ∝ Z^n, we can equate the exponents:

3/5 = n

Therefore, the value of n is 3/5.

**Explanation**

The given relationships describe how X, Y, and Z are related to each other. X varies as the cube of Y, which means as Y increases, X increases at a much faster rate. Similarly, Y varies as the 5th root of Z, which means as Z increases, Y also increases, but at a slower rate compared to X.

By combining these relationships, we can determine the relationship between X and Z. We find that X varies as Z^(3/5), indicating that as Z increases, X increases, but not as rapidly as Y increases.

The value of n is determined by equating the exponents in the equation X ∝ Z^n. In this case, the exponent is 3/5, so n = 3/5. This means that X varies as the 3/5th power of Z.

Overall, the relationships between X, Y, and Z can be understood by considering how the variables change in relation to each other. By analyzing the given information and using mathematical expressions, we can determine the value of n and understand the relationship between these variables.

N is 3/5