Determine the conjugate and the reciprocal of each complex number given below:
`i^(3)`

Answer

To determine the conjugate and the reciprocal of the complex number \( i^3 \), we will follow these steps:

### Step 1: Calculate \( i^3 \)
We know that \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \). The powers of \( i \) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

Thus, we can calculate:
\[ i^3 = -i \]

### Step 2: Find the Conjugate
The conjugate of a complex number \( a + bi \) is given by \( a - bi \). For our case:
- The complex number is \( 0 - i \) (which can be rewritten as \( 0 + (-1)i \)).
- Therefore, the conjugate of \( -i \) is:
\[ \text{Conjugate of } -i = 0 + i = i \]

### Step 3: Find the Reciprocal
The reciprocal of a complex number \( z = a + bi \) is given by:
\[ \frac{1}{z} = \frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2} \]
For \( z = -i \):
- Here, \( a = 0 \) and \( b = -1 \).
- Thus, \( a^2 + b^2 = 0^2 + (-1)^2 = 1 \).
- The reciprocal becomes:
\[ \frac{1}{-i} = \frac{0 + i}{1} = i \]

### Final Results
- The conjugate of \( i^3 \) (which is \( -i \)) is \( i \).
- The reciprocal of \( i^3 \) (which is \( -i \)) is \( i \).

### Summary
- Conjugate: \( i \)
- Reciprocal: \( i \)