The rate law for a reaction between the substances A and is given by Rate = [A]ⁿ [B]^m: On doubling the concentration of A and halving the concentration of B, the ratio of the new rate to the earlier rate of the reaction will be as:

\(2^{(n – m)}\)

The rate law for a reaction between the substances A and B is given by:
\[ \text{Rate} = [A]^n [B]^m \]

On doubling the concentration of A and halving the concentration of B, the ratio of the new rate to the earlier rate of the reaction will be as follows:

Doubling the concentration of A: \( [A] \rightarrow 2[A] \)

Halving the concentration of B: \( [B] \rightarrow 0.5[B] \)

Substituting these changes into the rate expression:

\[
\text{New Rate} = (2[A])^n (0.5[B])^m = 2^n [A]^n (0.5)^m [B]^m = 2^n (0.5)^m [A]^n [B]^m
\]

Therefore, the ratio of the new rate to the earlier rate is:

\[
\frac{\text{New Rate}}{\text{Earlier Rate}} = \frac{2^n (0.5)^m \cdot \text{Rate}}{\text{Rate}} = 2^n (0.5)^m
\]

Hence, the correct option would be (4) \( 2^{(n - m)} \), as it represents the ratio of the new rate to the earlier rate.