If `a^m .a^n = a^(mn)`, then `m(n - 2) + n(m- 2)` is :

Options

A. 0
B. 1
C. `-1`
D. `1/2`

Answer

To solve the problem, we need to simplify the expression \( m(n - 2) + n(m - 2) \) given the condition \( a^m \cdot a^n = a^{mn} \).

### Step-by-Step Solution:

1. **Start with the given expression**:
\[ m(n - 2) + n(m - 2) \]

2. **Distribute \( m \) and \( n \)**:
\[ = mn - 2m + nm - 2n \]
Here, we can see that \( mn \) and \( nm \) are the same, so we can combine them.

3. **Combine like terms**:
\[ = mn + nm - 2m - 2n \]
Since \( mn = nm \), we can rewrite this as:
\[ = 2mn - 2m - 2n \]

4. **Factor out the common factor**:
\[ = 2(mn - m - n) \]

5. **Substitute the relation from the given condition**:
From the condition \( a^m \cdot a^n = a^{m+n} \), we know that:
\[ mn = m + n \]
Therefore, we can substitute \( mn \) with \( m + n \):
\[ = 2((m + n) - m - n) \]

6. **Simplify the expression**:
\[ = 2(0) = 0 \]

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