` find multiplicative inverse of-i`

Explanation

To find the multiplicative inverse of \(-i\), we can follow these steps:

### Step 1: Define the multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by \( z^{-1} = \frac{1}{z} \). Here, we have \( z = -i \).

### Step 2: Write the expression for the multiplicative inverse
So, we need to find:
\[ z^{-1} = \frac{1}{-i} \]

### Step 3: Rationalize the denominator
To simplify \(\frac{1}{-i}\), we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(-i\) is \(i\):
\[ z^{-1} = \frac{1}{-i} \cdot \frac{i}{i} = \frac{i}{-i^2} \]

### Step 4: Simplify the expression
We know that \(i^2 = -1\), so:
\[ -i^2 = -(-1) = 1 \]
Thus, we have:
\[ z^{-1} = \frac{i}{1} = i \]

### Final Answer
The multiplicative inverse of \(-i\) is:
\[ i \]

---