Simplify the following Expression \( \frac{3+2 i}{1-2 i}=\square \)

To simplify the expression \( \frac{3+2 i}{1-2 i} \), we can multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 1 + 2 i \). Calculating the numerator: \[ (3 + 2i)(1 + 2i) = 3 \cdot 1 + 3 \cdot 2i + 2i \cdot 1 + 2i \cdot 2i = 3 + 6i + 2i + 4i^2 \] Since \( i^2 = -1 \), this becomes: \[ 3 + 6i + 2i - 4 = -1 + 8i \] Now for the denominator: \[ (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5 \] Now we can rewrite the expression: \[ \frac{3 + 2i}{1 - 2i} = \frac{-1 + 8i}{5} \] This simplifies to: \[ -\frac{1}{5} + \frac{8}{5} i \] Thus, the simplified form of the expression is: \[ \boxed{-\frac{1}{5} + \frac{8}{5} i} \]