`i^(57)+1/(i^(125))`, is equal to

Options

A. A. `2i`
B. B. `-2i`
C. C. `0`
D. D. `2`

Answer

To solve the expression \( i^{57} + \frac{1}{i^{125}} \), we can follow these steps:

### Step 1: Simplify \( i^{57} \)
We know that the powers of \( i \) (the imaginary unit) repeat every 4:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To simplify \( i^{57} \), we can find the remainder when 57 is divided by 4:
\[ 57 \div 4 = 14 \quad \text{(remainder 1)} \]
Thus, \( i^{57} = i^{4 \cdot 14 + 1} = (i^4)^{14} \cdot i^1 = 1^{14} \cdot i = i \).

### Step 2: Simplify \( \frac{1}{i^{125}} \)
Now, we simplify \( i^{125} \) in a similar way:
\[ 125 \div 4 = 31 \quad \text{(remainder 1)} \]
Thus, \( i^{125} = i^{4 \cdot 31 + 1} = (i^4)^{31} \cdot i^1 = 1^{31} \cdot i = i \).

So, \( \frac{1}{i^{125}} = \frac{1}{i} \).

### Step 3: Rationalize \( \frac{1}{i} \)
To rationalize \( \frac{1}{i} \), we multiply the numerator and denominator by \( -i \):
\[ \frac{1}{i} \cdot \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{1} = -i \]

### Step 4: Combine the results
Now we can combine the results from Step 1 and Step 3:
\[ i + \frac{1}{i^{125}} = i - i = 0 \]

### Final Answer
Thus, the expression \( i^{57} + \frac{1}{i^{125}} \) simplifies to:
\[ \boxed{0} \]