Determine the conjugate and the reciprocal of each complex number given below:
`sqrt(-9)-1`

Answer

To solve the problem of determining the conjugate and the reciprocal of the complex number \( \sqrt{-9} - 1 \), we can follow these steps:

### Step 1: Simplify the Complex Number
First, we need to simplify \( \sqrt{-9} - 1 \). We know that \( \sqrt{-9} \) can be expressed using the imaginary unit \( i \), where \( i^2 = -1 \).

\[ \sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i \]

Thus, we can rewrite the complex number:

\[ \sqrt{-9} - 1 = 3i - 1 \]

This can also be expressed in standard form as:

\[ -1 + 3i \]

### Step 2: Find the Conjugate
The conjugate of a complex number \( a + bi \) is given by \( a - bi \). For our complex number \( -1 + 3i \):

\[ \text{Conjugate} = -1 - 3i \]

### Step 3: Find the Reciprocal
To find the reciprocal of a complex number \( z = a + bi \), we use the formula:

\[ \frac{1}{z} = \frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2} \]

For our complex number \( -1 + 3i \):

- Here, \( a = -1 \) and \( b = 3 \).
- We calculate \( a^2 + b^2 \):

\[ a^2 + b^2 = (-1)^2 + (3)^2 = 1 + 9 = 10 \]

Now, we can find the reciprocal:

\[ \frac{1}{-1 + 3i} = \frac{-1 - 3i}{10} \]

This simplifies to:

\[ \frac{-1}{10} - \frac{3}{10}i \]

### Final Answers
- **Conjugate**: \( -1 - 3i \)
- **Reciprocal**: \( \frac{-1}{10} - \frac{3}{10}i \)

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