If the units of `M` and `L` are increased three times, then the unit of energy will be increased by

Options

A. (a)`3` times
B. (b)`6`times
C. (c)`27` times
D. (d)`81` times

Answer

To solve the problem, we need to analyze how the changes in the units of mass (M) and length (L) affect the unit of energy.

### Step-by-Step Solution:

1. **Understand the dimensions of energy**:
Energy (E) can be expressed in terms of mass (M), length (L), and time (T). The dimensional formula for energy is:
\[ [E] = [M]^1 [L]^2 [T]^{-2} \]

2. **Define the new units**:
If the units of mass and length are increased three times, we can express the new units as:
\[ M' = 3M \quad \text{and} \quad L' = 3L \]
The unit of time remains unchanged, so:
\[ T' = T \]

3. **Substitute the new units into the energy formula**:
Now, we substitute the new values of mass and length into the energy formula:
\[ E' = (M')^1 (L')^2 (T')^{-2} \]
Substituting the new values gives:
\[ E' = (3M)^1 (3L)^2 (T)^{-2} \]

4. **Calculate the new energy**:
Expanding this expression:
\[ E' = 3M \cdot (3^2 L^2) \cdot (T^{-2}) = 3M \cdot 9L^2 \cdot (T^{-2}) = 27 \cdot (M^1 L^2 T^{-2}) \]
Thus, we can express the new energy in terms of the original energy:
\[ E' = 27E \]

5. **Conclusion**:
Therefore, if the units of mass and length are increased three times, the unit of energy will be increased by a factor of 27.

### Final Answer:
The unit of energy will be increased by **27 times**.