The inverse of $ z $ is:
$$ z^{-1} = \frac{ 1125 }{ 62506 }+\frac{ 545 }{ 62506 }i $$We will denote the inverse of $ z $ as $ z_1 $. The inverse can be found in three steps.
Step 1: Rewrite complex number as its reciprocal
$$ z_1 = \frac{1}{ 45-\frac{ 109 }{ 5 }i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ 45-\frac{ 109 }{ 5 }i } \cdot \frac{ 45+\frac{ 109 }{ 5 }i }{ 45+\frac{ 109 }{ 5 }i } $$Step 3: Simplify
$$ z_1 = \frac{ 45+\frac{ 109 }{ 5 }i }{ \frac{ 62506 }{ 25 } } $$$$ z_1 = \frac{ 45 }{ \frac{ 62506 }{ 25 } } + \frac{ \frac{ 109 }{ 5 } }{ \frac{ 62506 }{ 25 } } \cdot i$$$$ z_1 = \frac{ 1125 }{ 62506 }+\frac{ 545 }{ 62506 }i $$