If `Z=(A^2B^3)/(C^4)`, then the relative error in Z will be :

Options

A. `(DeltaA)/A+(DeltaB)/B+(DeltaC)/C`
B. `(2DeltaA)/A+(3DeltaB)/B-(4DeltaC)/C`
C. `(2DeltaA)/A+(3DeltaB)/B+(4DeltaC)/C`
D. `(DeltaA)/A+(DeltaB)/B-(DeltaC)/C`

Explanation

To find the relative error in \( Z = \frac{A^2 B^3}{C^4} \), we can use the formula for the propagation of errors in multiplication and division.

### Step-by-Step Solution:

1. **Identify the expression for Z**:
\[ Z = \frac{A^2 B^3}{C^4} \]

2. **Rewrite Z**:
We can express \( Z \) in terms of powers:
\[ Z = A^2 B^3 C^{-4} \]

3. **Use the formula for relative error**:
The relative error in a product or quotient can be calculated using the following formula:
\[ \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C} \]
where \( \Delta A, \Delta B, \Delta C \) are the absolute errors in \( A, B, C \) respectively.

4. **Apply the powers to the relative errors**:
Since \( Z \) involves powers of \( A, B, \) and \( C \), we need to multiply the relative errors by their respective powers:
\[ \frac{\Delta Z}{Z} = 2 \frac{\Delta A}{A} + 3 \frac{\Delta B}{B} + 4 \frac{\Delta C}{C} \]

5. **Final expression for relative error in Z**:
Thus, the relative error in \( Z \) is given by:
\[ \frac{\Delta Z}{Z} = 2 \frac{\Delta A}{A} + 3 \frac{\Delta B}{B} + 4 \frac{\Delta C}{C} \]

### Conclusion:
The relative error in \( Z \) is:
\[ \frac{\Delta Z}{Z} = 2 \frac{\Delta A}{A} + 3 \frac{\Delta B}{B} + 4 \frac{\Delta C}{C} \]