The inverse of $ z $ is:
$$ z^{-1} = \frac{ 17600 }{ 52683 }+\frac{ 50000 }{ 52683 }i $$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
$$ z_1 = \frac{1}{ \frac{ 33 }{ 100 }-\frac{ 15 }{ 16 }i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ \frac{ 33 }{ 100 }-\frac{ 15 }{ 16 }i } \cdot \frac{ \frac{ 33 }{ 100 }+\frac{ 15 }{ 16 }i }{ \frac{ 33 }{ 100 }+\frac{ 15 }{ 16 }i } $$Step 3: Simplify
$$ z_1 = \frac{ \frac{ 33 }{ 100 }+\frac{ 15 }{ 16 }i }{ \frac{ 158049 }{ 160000 } } $$$$ z_1 = \frac{ \frac{ 33 }{ 100 } }{ \frac{ 158049 }{ 160000 } } + \frac{ \frac{ 15 }{ 16 } }{ \frac{ 158049 }{ 160000 } } \cdot i$$$$ z_1 = \frac{ 17600 }{ 52683 }+\frac{ 50000 }{ 52683 }i $$