Solve the followings :
`(a^(m-n))^(1) xx (a^(n-1))^m xx (a^(1 - m))^(n)`

Options

A. `1`
B. 0
C. 2
D. `a^(mn)`

Answer

To solve the expression \((a^{(m-n)})^{1} \times (a^{(n-1)})^{m} \times (a^{(1 - m)})^{n}\), we will follow these steps:

### Step 1: Simplify Each Term
First, we can simplify each term in the expression:
- The first term \((a^{(m-n)})^{1}\) simplifies to \(a^{(m-n)}\).
- The second term \((a^{(n-1)})^{m}\) simplifies to \(a^{(m(n-1))} = a^{(mn - m)}\).
- The third term \((a^{(1-m)})^{n}\) simplifies to \(a^{(n(1-m))} = a^{(n - nm)}\).

So, we rewrite the expression as:
\[ a^{(m-n)} \times a^{(mn - m)} \times a^{(n - nm)} \]

### Step 2: Combine the Exponents
Since the bases are the same (all are \(a\)), we can add the exponents:
\[ (m - n) + (mn - m) + (n - nm) \]

### Step 3: Simplify the Exponent
Now, let's simplify the exponent:
1. Start with the expression:
\[ (m - n) + (mn - m) + (n - nm) \]
2. Combine like terms:
- The \(m\) terms: \(m - m\) cancels out.
- The \(n\) terms: \(-n + n\) cancels out.
- The remaining terms are \(mn - nm\), which is \(0\).

Thus, we have:
\[ 0 \]

### Step 4: Final Result
Now, we can express the final result:
\[ a^{0} = 1 \]

So, the final answer is:
\[ 1 \]