Evaluate ` [ i^(14) - (1/i)^(34)]^2`

Answer

To evaluate the expression \( [ i^{14} - (1/i)^{34}]^2 \), we will follow these steps:

### Step 1: Simplify \( i^{14} \)
We know that \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \). The powers of \( i \) cycle every 4:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To find \( i^{14} \), we can reduce the exponent modulo 4:
\[ 14 \mod 4 = 2 \]
Thus,
\[ i^{14} = i^2 = -1 \]

### Step 2: Simplify \( (1/i)^{34} \)
We can rewrite \( \frac{1}{i} \) as \( -i \) (since \( \frac{1}{i} \cdot i = 1 \)). Therefore,
\[ (1/i)^{34} = (-i)^{34} \]
Now, we simplify \( (-i)^{34} \). The powers of \( -i \) also cycle every 4:
- \( (-i)^1 = -i \)
- \( (-i)^2 = -1 \)
- \( (-i)^3 = i \)
- \( (-i)^4 = 1 \)

Now, we find \( 34 \mod 4 \):
\[ 34 \mod 4 = 2 \]
Thus,
\[ (-i)^{34} = (-i)^2 = -1 \]

### Step 3: Substitute back into the expression
Now we substitute back into our original expression:
\[ i^{14} - (1/i)^{34} = -1 - (-1) = -1 + 1 = 0 \]

### Step 4: Square the result
Now we square the result:
\[ [ i^{14} - (1/i)^{34}]^2 = 0^2 = 0 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{0} \]