For a positive integer n , find the value of `(1-i)^(n) (1 - 1/i)^(n)`

Answer

To solve the expression \((1-i)^{n} \cdot (1 - \frac{1}{i})^{n}\) for a positive integer \(n\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1-i)^{n} \cdot \left(1 - \frac{1}{i}\right)^{n} \]

### Step 2: Simplify \(1 - \frac{1}{i}\)
To simplify \(1 - \frac{1}{i}\), we can multiply and divide by \(i\):
\[ 1 - \frac{1}{i} = 1 - \left(-i\right) = 1 + i \]
Thus, we have:
\[ \left(1 - \frac{1}{i}\right) = 1 + i \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our expression:
\[ (1-i)^{n} \cdot (1+i)^{n} \]

### Step 4: Combine the terms
We can combine the terms:
\[ ((1-i)(1+i))^{n} \]

### Step 5: Use the difference of squares
Using the identity \(a^2 - b^2 = (a-b)(a+b)\), we have:
\[ (1-i)(1+i) = 1^2 - i^2 \]
Since \(i^2 = -1\), we can simplify this to:
\[ 1 - (-1) = 1 + 1 = 2 \]

### Step 6: Raise to the power of \(n\)
Now, substituting this back, we have:
\[ (2)^{n} \]

### Final Answer
Thus, the final value of the expression \((1-i)^{n} \cdot (1 - \frac{1}{i})^{n}\) is:
\[ 2^{n} \]

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