The inverse of $ z $ is:
$$ z^{-1} = \frac{ 4 }{ 25 }-\frac{ 3 }{ 25 }i $$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
$$ z_1 = \frac{1}{ 4+3i } $$Step 2: Multiply top and bottom by complex conjugate of $ z $
$$ z_1 = \frac{1}{ 4+3i } \cdot \frac{ 4-3i }{ 4-3i } $$Step 3: Simplify
$$ z_1 = \frac{ 4-3i }{ 25 } $$$$ z_1 = \frac{ 4 }{ 25 } - \frac{ 3 }{ 25 } \cdot i$$$$ z_1 = \frac{ 4 }{ 25 }-\frac{ 3 }{ 25 }i $$