# How do you simplify (2+3i)/(1+2i)?

May 21, 2018

$\frac{8 - i}{5}$

#### Explanation:

$\frac{2 + 3 i}{1 + 2 i}$

You multiply it by the complex conjugate to get rid of the imaginary numbers in the denominator, since the conjugate's value is 1 the value of the expression is not altered:

complex conjugate is the $\text{denominator"/"denominator}$ with the sign changed:

$\frac{1 - 2 i}{1 - 2 i} = 1$

$\frac{2 + 3 i}{1 + 2 i} \cdot \frac{1 - 2 i}{1 - 2 i}$

$= \frac{2 - 4 i + 3 i - 6 {i}^{2}}{1 - 2 i + 2 i - 4 {i}^{2}}$

$= \frac{2 - i - 6 {\sqrt{- 1}}^{2}}{1 - 4 {\sqrt{- 1}}^{2}}$

$= \frac{2 - i - 6 \cdot - 1}{1 - 4 \cdot - 1}$

$= \frac{2 - i + 6}{1 + 4}$

$= \frac{8 - i}{5}$