# How do you simplify (2-3i) div (1+5i)?

Jan 2, 2016

Multiply dividend and divisor by the Complex conjugate of the divisor then simplify to find:

$\left(2 - 3 i\right) \div \left(1 + 5 i\right) = - \frac{1}{2} - \frac{1}{2} i$

#### Explanation:

The Complex conjugate of the divisor is $\left(1 - 5 i\right)$.

If we multiply both the dividend and the divisor by $\left(1 - 5 i\right)$ and simplify then we will get a Real divisor without altering the quotient...

$\frac{2 - 3 i}{1 + 5 i}$

$= \frac{\left(2 - 3 i\right) \left(1 - 5 i\right)}{\left(1 + 5 i\right) \left(1 - 5 i\right)}$

$= \frac{2 - 10 i - 3 i + 15 {i}^{2}}{{1}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{5 i}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{5 i}}} - {\left(5 i\right)}^{2}}$

$= \frac{- 13 - 13 i}{1 + 25}$

$= \frac{- 13 \left(1 + i\right)}{26}$

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