Two physical quantities A and B have different dimensions. Which mathematical operation given below is physically possible ?

Options

A. `sqrt(AB)`
B. `A(1+B)`
C. `A-B`
D. `A+B`

Answer

To determine which mathematical operation between two physical quantities A and B with different dimensions is physically possible, we can analyze the operations based on dimensional analysis principles. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Operations
We need to evaluate the mathematical operations involving A and B. The operations we will consider are:
- Addition (A + B)
- Subtraction (A - B)
- Multiplication (A × B)
- Division (A / B)
- Raising to a power (A^n or B^n)

### Step 2: Analyze Addition and Subtraction
According to dimensional analysis, two physical quantities can only be added or subtracted if they have the same dimensions. Since A and B have different dimensions, the operations of addition (A + B) and subtraction (A - B) are not physically possible.

**Hint:** Remember that only quantities with the same dimensions can be combined through addition or subtraction.

### Step 3: Analyze Multiplication
Multiplication of two physical quantities is always possible, regardless of their dimensions. When we multiply A and B, we get a new quantity with a dimension that is a product of the dimensions of A and B.

**Hint:** Multiplication of different dimensional quantities results in a new dimension, which is always valid.

### Step 4: Analyze Division
Similarly, division of two physical quantities is also possible, regardless of their dimensions. Dividing A by B results in a new dimension that is the ratio of the dimensions of A and B.

**Hint:** Division of quantities with different dimensions yields a new dimension, making it a valid operation.

### Step 5: Analyze Raising to a Power
Raising a quantity to a power is also valid. For example, A^1/2 (the square root of A) or B^1/2 (the square root of B) is valid. However, if we consider an operation like A × B^(1/2), this is also valid since we are multiplying A by a function of B.

**Hint:** Raising a quantity to a power does not depend on the dimensions being the same.

### Step 6: Conclusion
From the analysis:
- Addition and subtraction are not possible due to different dimensions.
- Multiplication and division are possible regardless of dimensions.
- Raising to a power is also valid.

Thus, the only mathematically feasible operation among the options provided is the multiplication of A and B raised to a power (e.g., A × B^(1/2)).

### Final Answer
The mathematically possible operation is:
**A × B^(1/2)** (Option A).