Let's solve the problem step by step.

Given:

- ${7056}_n-{4577}_n={2257}_n$

We need to find the value of the base $n$.

First, let's convert the numbers from base $n$ to base 10:

1. ${7056}_n$ in base 10:
$$7n^3+0n^2+5n+6$$

2. ${4577}_n$ in base 10:
$$4n^3+5n^2+7n+7$$

3. ${2257}_n$ in base 10:
$$2n^3+2n^2+5n+7$$

The equation becomes:

$$(7n^3+0n^2+5n+6)-(4n^3+5n^2+7n+7)=2n^3+2n^2+5n+7$$

Simplifying this:

$$7n^3+5n+6-4n^3-5n^2-7n-7=2n^3+2n^2+5n+7$$

Combine like terms:

$$3n^3-5n^2-2n-1=2n^3+2n^2+5n+7$$

Bring all terms to one side:

$$3n^3-5n^2-2n-1-2n^3-2n^2-5n-7=0$$

Simplify:

$$n^3-7n^2-7n-8=0$$

To find the value of $n$, substitute the given options one by one:

1. $n=8\frac{17}{30}$:

$$n=\frac{30\times 8+17}{30}=\frac{240+17}{30}=\frac{257}{30}$$

Check if the base $n=8.5667$:


Answer: A. $8\frac{17}{30}$