If dimensions of A and B are different, then which of the following operaion is valid ?

Options

A. `(A)/(B)`
B. `e^(-A//B)`
C. A-B
D. A+B

Answer

To determine which operation is valid when the dimensions of A and B are different, we need to analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understanding Dimensions**:
- Dimensions are a way to express physical quantities in terms of basic quantities such as mass (M), length (L), time (T), etc. If two quantities have different dimensions, they cannot be directly added, subtracted, or exponentiated in certain ways.

2. **Analyzing Each Operation**:
- **Option 1: A/B**
- In this operation, we are dividing A by B. The result of this operation will have its own dimension, which is derived from the dimensions of A and B. Since division is allowed between different dimensions, this operation is valid regardless of whether A and B have the same dimensions or not.

- **Option 2: e^(-A/B)**
- The expression e^(-A/B) involves an exponent. For exponentiation to be valid, the argument (in this case, -A/B) must be dimensionless. This means that A and B must have the same dimensions for the division A/B to yield a dimensionless quantity. Since we are given that A and B have different dimensions, this operation is not valid.

- **Option 3: A - B**
- Subtraction requires that the quantities being subtracted have the same dimensions. Since A and B have different dimensions, this operation is also not valid.

- **Option 4: A + B**
- Similar to subtraction, addition also requires that the quantities being added have the same dimensions. Since A and B have different dimensions, this operation is not valid.

3. **Conclusion**:
- The only valid operation among the options provided is **A/B**. Therefore, the correct answer is **Option 1**.

### Final Answer:
The valid operation when the dimensions of A and B are different is **A/B**.