For `10^((at+3))`, the dimension of a is -

Options

A. `M^(0)L^(0)T^(0)`
B. `M^(0)L^(0)T^(1)`
C. `M^(0)L^(0)T^(-1)`
D. none of these

Answer

To determine the dimension of \( a \) in the expression \( 10^{(at + 3)} \), we can follow these steps:

### Step 1: Understand the expression
The expression \( 10^{(at + 3)} \) involves an exponent. In general, for any base \( b \) raised to a power, the power must be dimensionless. This means that the entire exponent \( at + 3 \) must not have any dimensions.

### Step 2: Analyze the components of the exponent
In the exponent \( at + 3 \):
- The term \( 3 \) is a constant and is dimensionless.
- The term \( at \) must also be dimensionless for the entire exponent to be dimensionless.

### Step 3: Identify the dimensions of \( t \)
Let \( t \) represent time. The dimension of time is denoted as \( [T] \).

### Step 4: Determine the dimension of \( a \)
Since \( at \) must be dimensionless, we can express this relationship mathematically:
\[ [a][T] = 1 \]
This implies that the dimension of \( a \) must be the inverse of the dimension of time:
\[ [a] = [T]^{-1} \]

### Step 5: Conclusion
Thus, the dimension of \( a \) is \( T^{-1} \), which is the same as saying \( a \) has the dimension of frequency.

### Final Answer
The dimension of \( a \) is \( T^{-1} \).

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