# How do you simplify (2-i)/(3-4i)?

Dec 20, 2016

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

#### Explanation:

To simplify the fraction we require to have a rational denominator.

This is achieved by multiplying the numerator/denominator by the $\textcolor{b l u e}{\text{complex conjugate}}$ of the complex number on the denominator.

$\text{The conjugate of " 3-4i" is } 3 + 4 i$

$\Rightarrow \frac{2 - i}{3 - 4 i} = \frac{\left(2 - i\right) \left(3 + 4 i\right)}{\left(3 - 4 i\right) \left(3 + 4 i\right)}$

distributing the numerator/denominator using the FOIL method.

$= \frac{6 + 8 i - 3 i - 4 {i}^{2}}{9 + 12 i - 12 i - 16 {i}^{2}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder: }} {i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1$

$= \frac{6 + 5 i + 4}{9 + 16} = \frac{10 + 5 i}{25} \leftarrow \text{ rational denominator}$

$\text{Expressing in " color(blue)"standard form}$

$= \frac{10}{25} + \frac{5}{25} i = \frac{2}{5} + \frac{1}{5} i$

Dec 20, 2016

$\frac{2 + i}{5}$.

#### Explanation:

Multiply by the complex conjugate of the denominator, in this case $3 + 4 i$.

$\frac{2 - i}{3 - 4 i} = \frac{2 - i}{3 - 4 i} \cdot \frac{3 + 4 i}{3 + 4 i} = \frac{\left(2 - i\right) \left(3 + 4 i\right)}{\left(3 - 4 i\right) \left(3 + 4 i\right)}$.

Numerator: $\left(2 - i\right) \left(3 + 4 i\right) = 6 + 8 i - 3 i - 4 {i}^{2} = 6 + 5 i + 4 = 10 + 5 i$.

Denominator: $\left(3 - 4 i\right) \left(3 + 4 i\right) = 9 + 12 i - 12 i - 16 {i}^{2} = 9 + 16 = 25$.

So we have $\frac{\left(2 - i\right) \left(3 + 4 i\right)}{\left(3 - 4 i\right) \left(3 + 4 i\right)} = \frac{10 + 5 i}{25} = \frac{2 + i}{5}$.