The value of ` i^(-9999)` is

Options

A. `-i`
B. `i`
C. `1`
D. `-1`

Answer

To find the value of \( i^{-9999} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ i^{-9999} = \frac{1}{i^{9999}} \]

**Hint**: Remember that negative exponents indicate the reciprocal of the base raised to the positive exponent.

### Step 2: Determine \( 9999 \mod 4 \)
Next, we need to find \( 9999 \mod 4 \) to simplify \( i^{9999} \). We divide \( 9999 \) by \( 4 \):
\[ 9999 \div 4 = 2499 \quad \text{(remainder 3)} \]
Thus, \( 9999 = 4 \times 2499 + 3 \), which means:
\[ 9999 \mod 4 = 3 \]

**Hint**: When dividing by 4, the remainder will help us determine the equivalent power of \( i \).

### Step 3: Use the cyclic nature of \( i \)
From the properties of \( i \):
- \( i^0 = 1 \)
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \) (and it repeats every 4)

Since \( 9999 \mod 4 = 3 \), we have:
\[ i^{9999} = i^3 = -i \]

**Hint**: The powers of \( i \) repeat every four terms, so knowing the remainder when divided by 4 is crucial.

### Step 4: Substitute back into the expression
Now we can substitute back into our expression for \( i^{-9999} \):
\[ i^{-9999} = \frac{1}{i^{9999}} = \frac{1}{-i} \]

**Hint**: We are now working with the reciprocal of a complex number.

### Step 5: Multiply numerator and denominator by \( i \)
To simplify \( \frac{1}{-i} \), we multiply the numerator and denominator by \( i \):
\[ \frac{1}{-i} \cdot \frac{i}{i} = \frac{i}{-i^2} = \frac{i}{-(-1)} = \frac{i}{1} = i \]

**Hint**: Multiplying by the complex conjugate can help eliminate imaginary units in the denominator.

### Final Answer
Thus, the value of \( i^{-9999} \) is:
\[ \boxed{i} \]