The inverse of $ z $ is:
$$ z^{-1} = -\frac{ 2 }{ 15 }-\frac{ 1 }{ 15 }i $$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
$$ z_1 = \frac{1}{ -6+3i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ -6+3i } \cdot \frac{ -6-3i }{ -6-3i } $$Step 3: Simplify
$$ z_1 = \frac{ -6-3i }{ 45 } $$$$ z_1 = \frac{ -6 }{ 45 } - \frac{ 3 }{ 45 } \cdot i$$$$ z_1 = -\frac{ 2 }{ 15 }-\frac{ 1 }{ 15 }i $$