The inverse of $ z $ is:
$$ z^{-1} = \frac{ 205 }{ 1706 }-\frac{ 25 }{ 1706 }i $$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
$$ z_1 = \frac{1}{ \frac{ 41 }{ 5 }+i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ \frac{ 41 }{ 5 }+i } \cdot \frac{ \frac{ 41 }{ 5 }-i }{ \frac{ 41 }{ 5 }-i } $$Step 3: Simplify
$$ z_1 = \frac{ \frac{ 41 }{ 5 }-i }{ \frac{ 1706 }{ 25 } } $$$$ z_1 = \frac{ \frac{ 41 }{ 5 } }{ \frac{ 1706 }{ 25 } } - \frac{ 1 }{ \frac{ 1706 }{ 25 } } \cdot i$$$$ z_1 = \frac{ 205 }{ 1706 }-\frac{ 25 }{ 1706 }i $$