The dimensional formula of 'b' is not the same as the dimensional formula of pressure (P), Boltzmann's constant (K), temperature (T), or time (t).

To understand why, let's analyze the given formula and try to determine the dimensions of each variable.

The formula given is: P = a * exp(-b * t)

- Pressure (P): The dimensional formula of pressure is [M L^-1 T^-2], where M represents mass, L represents length, and T represents time.
- Boltzmann's constant (K): The dimensional formula of Boltzmann's constant is [M L^2 T^-2 θ^-1], where θ represents temperature in Kelvin.
- Temperature (T): The dimensional formula of temperature is [θ], where θ represents temperature in Kelvin.
- Time (t): The dimensional formula of time is [T], where T represents time.

Now, let's determine the dimensional formula of 'b' by comparing the dimensions of both sides of the equation:

On the left side, we have the dimensional formula of pressure: [M L^-1 T^-2]
On the right side, we have the dimensional formula of 'a * exp(-b * t)'

Since 'a' is a constant, its dimensional formula is just a number and does not affect the dimensional formula of 'b'.

The exponential function exp(-b * t) has a dimensionless argument, as the exponential function is a pure number.

Therefore, the dimensional formula of 'b' can be determined by subtracting the dimensions of pressure from the dimensions of the exponential function:

[M L^-1 T^-2] - [dimensionless] = [M L^-1 T^-2]

Hence, the dimensional formula of 'b' is the same as the dimensional formula of pressure (P).

In summary, the dimensional formula of 'b' is not the same as the dimensional formula of temperature (T), Boltzmann's constant (K), or time (t). The dimensional formula of 'b' is the same as the dimensional formula of pressure (P).