Which of the following functions of A and B may be performed if A and B posses different dimensions

Options

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

Answer

To solve the question regarding which functions of A and B can be performed if A and B possess different dimensions, we will analyze the operations one by one based on the principle of homogeneity.

### Step-by-Step Solution:

1. **Understanding Dimensions**:
- A and B have different dimensions. For example, let A be a length (meters) and B be a time (seconds).

2. **Addition and Subtraction**:
- According to the principle of homogeneity, quantities with different dimensions cannot be added or subtracted.
- Therefore, operations such as A + B or A - B are not permissible.

3. **Multiplication**:
- When multiplying two quantities with different dimensions, the result will have a new dimension.
- For example, if A is a length (L) and B is a force (F), then A × B will yield a quantity with the dimension of work (W), which is expressed as force × distance (F × L).

4. **Division**:
- Similarly, dividing two quantities with different dimensions also results in a new dimension.
- For instance, if A is distance (L) and B is time (T), then A / B will yield a quantity with the dimension of velocity (V), which is L/T.

5. **Conclusion**:
- Based on the analysis, we conclude that:
- **Addition and subtraction** of A and B cannot be performed.
- **Multiplication and division** of A and B can be performed, resulting in new dimensions.

### Final Answer:
- The functions that may be performed if A and B possess different dimensions are multiplication and division.