If A and B are two physical quantities having different dimensions then which of the following can denote a new physical quantity?

Options

A. `A+A^(3)/B`
B. `exp (-A/B)`
C. `AB^(2)`
D. `A/B^(4)`

Explanation

To solve the problem of determining which of the given expressions can denote a new physical quantity when A and B have different dimensions, we will analyze each option step-by-step.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have two physical quantities, A and B, which have different dimensions. We need to check which of the given options can represent a new physical quantity.

2. **Option 1: A + (A³ / B)**
- For addition (A + B) to be valid, both terms must have the same dimensions.
- The dimension of A is [A], and the dimension of (A³ / B) is [A³] / [B].
- Since A and B have different dimensions, [A] ≠ [B], thus [A] ≠ [A³ / B].
- Therefore, this expression does not yield a new dimension.

3. **Option 2: e^(-A/B)**
- The expression involves an exponential function.
- The exponent of any number must be dimensionless.
- The dimension of (-A/B) is [A] / [B], which is not dimensionless since A and B have different dimensions.
- Hence, this expression does not yield a new dimension.

4. **Option 3: AB²**
- The dimension of this expression can be calculated as follows:
- Dimension of A = [A]
- Dimension of B = [B]
- Therefore, the dimension of AB² = [A] * [B]² = [A][B²].
- This is a new dimension that can be derived from A and B.

5. **Option 4: A / B⁴**
- The dimension of this expression can be calculated as follows:
- Dimension of A = [A]
- Dimension of B = [B]
- Therefore, the dimension of A / B⁴ = [A] / [B]⁴ = [A][B]⁻⁴.
- This also represents a new dimension derived from A and B.

### Conclusion:
- The expressions that can denote a new physical quantity are:
- Option 3: AB²
- Option 4: A / B⁴