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+3B
B. 3A-B
C. `A^3+(3B)^3`
D. `A^(1//3)xx(3B)^3`

Answer

To determine which of the given options can denote a new physical quantity when A and B are two physical quantities with different dimensions, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Dimensions**:
- A and B have different dimensions. This means that they cannot be directly added or subtracted because physical quantities must have the same dimensions to perform such operations.

2. **Analyzing Option A: A + 3B**:
- The expression A + 3B combines A and B.
- Since A and B have different dimensions, the sum A + 3B does not yield a new physical quantity.
- **Conclusion**: Option A does not denote a new physical quantity.

3. **Analyzing Option B: 3A - B**:
- Similar to Option A, this expression also combines A and B.
- The subtraction of B from 3A does not change the fact that A and B have different dimensions.
- **Conclusion**: Option B does not denote a new physical quantity.

4. **Analyzing Option C: A³ + 3B³**:
- In this case, we are cubing both A and B.
- Although both terms are raised to a power, they still have different dimensions.
- The addition of A³ and 3B³ does not create a new physical quantity because they remain incompatible in terms of dimensions.
- **Conclusion**: Option C does not denote a new physical quantity.

5. **Analyzing Option D: (A^(1/3)) * (3B³)**:
- Here, A is raised to the power of 1/3 and B is raised to the power of 3.
- This operation combines A and B in a way that alters their dimensions.
- The dimensions of A^(1/3) and B³ can potentially combine to form a new physical quantity because they are not directly added or subtracted.
- **Conclusion**: Option D can denote a new physical quantity.

### Final Answer:
The option that can denote a new physical quantity is **Option D**.