Find the multiplicative inverse of each of the following complex numbers when it exists.
`-7+ 0i`

Answer

To find the multiplicative inverse of the complex number \(-7 + 0i\), we follow these steps:

### Step 1: Identify the complex number
The complex number given is:
\[ z = -7 + 0i \]

### Step 2: Write the formula for the multiplicative inverse
The multiplicative inverse of a complex number \(z\) is given by:
\[ \frac{1}{z} \]

### Step 3: Substitute the complex number into the formula
Substituting the value of \(z\):
\[ \frac{1}{-7 + 0i} \]

### Step 4: Simplify the expression
To simplify \(\frac{1}{-7 + 0i}\), we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(-7 + 0i\) is \(-7 - 0i\) (which is simply \(-7\)):
\[ \frac{1 \cdot (-7)}{(-7)(-7) + (0)(0)} = \frac{-7}{49} = -\frac{1}{7} \]

### Step 5: Determine if the multiplicative inverse exists
The multiplicative inverse exists as long as the complex number is not equal to zero. Since \(-7 + 0i\) is not zero, the multiplicative inverse exists.

### Conclusion
Thus, the multiplicative inverse of the complex number \(-7 + 0i\) is:
\[ -\frac{1}{7} \]

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