When `a^m cdot a^n = a^(mn)`,find the value of n(m-2) + m(n -2) is :

Options

A. 0
B. 1
C. -1
D. none of these

Answer

To solve the problem, we need to analyze the given equation and simplify the expression step by step.

### Step 1: Understand the Given Equation
We are given that:
\[ a^m \cdot a^n = a^{mn} \]

Using the property of exponents, we know that:
\[ a^m \cdot a^n = a^{m+n} \]

So, we can set up the equation:
\[ a^{m+n} = a^{mn} \]

### Step 2: Set the Exponents Equal
Since the bases are the same (both are \(a\)), we can equate the exponents:
\[ m + n = mn \]

### Step 3: Rearranging the Equation
Rearranging the equation gives us:
\[ mn - m - n = 0 \]

### Step 4: Finding the Expression
We need to find the value of:
\[ n(m-2) + m(n-2) \]

### Step 5: Expand the Expression
Let's expand this expression:
\[ n(m-2) + m(n-2) = nm - 2n + mn - 2m \]
Combining like terms:
\[ = 2mn - 2n - 2m \]

### Step 6: Factor Out Common Terms
We can factor out 2 from the expression:
\[ = 2(mn - n - m) \]

### Step 7: Substitute from the Rearranged Equation
From our earlier rearranged equation \(mn - m - n = 0\), we can substitute:
\[ mn - n - m = 0 \]
Thus:
\[ = 2(0) = 0 \]

### Conclusion
The value of \( n(m-2) + m(n-2) \) is:
\[ \boxed{0} \]