The inverse of $ z $ is:
$$ z^{-1} = \frac{ 3 }{ 4 }- \frac{\sqrt{ 3 }}{ 4 }i $$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
$$ z_1 = \frac{1}{ 1+\frac{\sqrt{ 3 }}{ 3 }i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ 1+\frac{\sqrt{ 3 }}{ 3 }i } \cdot \frac{ 1- \frac{\sqrt{ 3 }}{ 3 }i }{ 1- \frac{\sqrt{ 3 }}{ 3 }i } $$Step 3: Simplify
$$ z_1 = \frac{ 1- \frac{\sqrt{ 3 }}{ 3 }i }{ \frac{ 4 }{ 3 } } $$$$ z_1 = \frac{ 1 }{ \frac{ 4 }{ 3 } } - \frac{ \frac{\sqrt{ 3 }}{ 3 } }{ \frac{ 4 }{ 3 } } \cdot i$$$$ z_1 = \frac{ 3 }{ 4 }- \frac{\sqrt{ 3 }}{ 4 }i $$