If `a^(m).a^(n)=a^(nm)`, then find the value of `m(n-2)+n(m-2)`.

Answer

To solve the problem step by step, we start with the given equation and follow through with the necessary algebraic manipulations.

### Step-by-Step Solution:

1. **Understand the Given Equation:**
We are given that \( a^m \cdot a^n = a^{mn} \). According to the laws of exponents, when multiplying two powers with the same base, we add the exponents. Therefore, we can rewrite this as:
\[ a^{m+n} = a^{mn} \]
This implies:
\[ m + n = mn \]

2. **Write the Expression to be Evaluated:**
We need to find the value of the expression:
\[ m(n-2) + n(m-2) \]

3. **Expand the Expression:**
Let's expand the expression:
\[ m(n-2) + n(m-2) = mn - 2m + nm - 2n \]
Since \( mn \) and \( nm \) are the same, we can combine them:
\[ = 2mn - 2m - 2n \]

4. **Substitute \( mn \) with \( m+n \):**
From our earlier step, we know that \( mn = m + n \). We substitute this into our expression:
\[ = 2(m+n) - 2m - 2n \]

5. **Simplify the Expression:**
Now, simplify the expression:
\[ = 2m + 2n - 2m - 2n \]
All terms cancel out:
\[ = 0 \]

### Final Answer:
Thus, the value of \( m(n-2) + n(m-2) \) is:
\[ \boxed{0} \]