FInd the value of `\ (1+i)^(4) xx (1 + (1)/(i))^(4)`

Options

A. `1`
B. `16`
C. `0`
D. `-1`

Explanation

To solve the problem \( (1+i)^{4} \times \left(1 + \frac{1}{i}\right)^{4} \), we will break it down into manageable steps.

### Step 1: Simplify \( \left(1 + \frac{1}{i}\right) \)

First, we need to simplify \( 1 + \frac{1}{i} \). To do this, we multiply the numerator and the denominator by \( i \):

\[ 1 + \frac{1}{i} = 1 + \frac{1 \cdot i}{i \cdot i} = 1 + \frac{i}{-1} = 1 - i \]

### Step 2: Calculate \( (1+i)^{4} \)

Next, we calculate \( (1+i)^{4} \). We can use the binomial theorem or directly calculate it step by step.

First, calculate \( (1+i)^{2} \):

\[ (1+i)^{2} = 1^{2} + 2 \cdot 1 \cdot i + i^{2} = 1 + 2i - 1 = 2i \]

Now, raise \( 2i \) to the power of 2 to find \( (1+i)^{4} \):

\[ (2i)^{2} = 4i^{2} = 4(-1) = -4 \]

### Step 3: Calculate \( (1-i)^{4} \)

Now, we calculate \( (1-i)^{4} \). Similar to the previous step, we first find \( (1-i)^{2} \):

\[ (1-i)^{2} = 1^{2} - 2 \cdot 1 \cdot i + i^{2} = 1 - 2i - 1 = -2i \]

Now, raise \( -2i \) to the power of 2 to find \( (1-i)^{4} \):

\[ (-2i)^{2} = 4i^{2} = 4(-1) = -4 \]

### Step 4: Combine the results

Now we can combine the results from Step 2 and Step 3:

\[ (1+i)^{4} \times (1-i)^{4} = (-4) \times (-4) = 16 \]

### Final Answer

Thus, the value of \( (1+i)^{4} \times \left(1 + \frac{1}{i}\right)^{4} \) is:

\[ \boxed{16} \]