Find the multiplicative inverse of each of the following complex numbers when it exists.
`2 + 2i`

Answer

To find the multiplicative inverse of the complex number \( z = 2 + 2i \), we will follow these steps:

### Step 1: Write the multiplicative inverse formula
The multiplicative inverse of a complex number \( z \) is given by:
\[ \frac{1}{z} \]
For our case, we have:
\[ \frac{1}{2 + 2i} \]

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

### Step 3: Simplify the denominator
Now we need to simplify the denominator:
\[ (2 + 2i)(2 - 2i) = 2^2 - (2i)^2 = 4 - 4(-1) = 4 + 4 = 8 \]
So, the expression becomes:
\[ \frac{2 - 2i}{8} \]

### Step 4: Simplify the fraction
Now we can simplify the fraction:
\[ \frac{2 - 2i}{8} = \frac{2}{8} - \frac{2i}{8} = \frac{1}{4} - \frac{1}{4}i \]

### Step 5: Write in standard form
The multiplicative inverse of \( 2 + 2i \) in the standard form \( a + bi \) is:
\[ \frac{1}{4} - \frac{1}{4}i \]

### Final Answer
Thus, the multiplicative inverse of \( 2 + 2i \) is:
\[ \frac{1}{4} - \frac{1}{4}i \]

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