If `z=i^(-39)`, then simplest form of z is equal to a +i. The value of a is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem where \( z = i^{-39} \) and express it in the form \( a + i \), we will follow these steps:

### Step-by-Step Solution:

1. **Rewrite the expression**: Start with the given expression for \( z \):
\[ z = i^{-39} \]

2. **Use the property of exponents**: Recall that \( i^{-n} = \frac{1}{i^n} \). Therefore, we can rewrite \( z \):
\[ z = \frac{1}{i^{39}} \]

3. **Simplify \( i^{39} \)**: To simplify \( i^{39} \), we use the fact that the powers of \( i \) cycle every 4:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To find \( i^{39} \), we calculate \( 39 \mod 4 \):
\[ 39 \div 4 = 9 \quad \text{(remainder 3)} \]
Thus, \( 39 \mod 4 = 3 \), and therefore:
\[ i^{39} = i^3 = -i \]

4. **Substitute back into the expression for \( z \)**:
\[ z = \frac{1}{-i} \]

5. **Multiply numerator and denominator by \( i \)** to eliminate the imaginary unit from the denominator:
\[ z = \frac{1 \cdot i}{-i \cdot i} = \frac{i}{-(-1)} = \frac{i}{1} = -i \]

6. **Express in the form \( a + i \)**: We can rewrite \( -i \) as:
\[ z = 0 + (-1)i \]
Here, we can see that \( a = 0 \).

### Conclusion:
The value of \( a \) is:
\[ \boxed{0} \]