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: Evaluate \( i^{14} \)

We know that the powers of \( i \) (the imaginary unit) 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: Evaluate \( (1/i)^{34} \)

We can rewrite \( \frac{1}{i} \) as \( -i \) (since \( \frac{1}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i \)). Therefore:
\[ (1/i)^{34} = (-i)^{34} \]

Now, we can reduce \( (-i)^{34} \) using the properties of powers:
\[ (-i)^{34} = (-1)^{34} \cdot i^{34} \]
Since \( (-1)^{34} = 1 \), we only need to evaluate \( i^{34} \):
\[ 34 \mod 4 = 2 \]
Thus, \( i^{34} = i^2 = -1 \), and therefore:
\[ (1/i)^{34} = 1 \cdot (-1) = -1 \]

### Step 3: Substitute values into the expression

Now we substitute the values we found back into the expression:
\[ i^{14} - (1/i)^{34} = -1 - (-1) = -1 + 1 = 0 \]

### Step 4: Square the result

Finally, we need to square the result:
\[ [ i^{14} - (1/i)^{34} ]^2 = [0]^2 = 0 \]

### Final Answer
The value of \( [ i^{14} - (1/i)^{34}]^2 \) is \( 0 \).

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