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

Mar 18, 2018

$\frac{8 + 12 i}{13}$

#### Explanation:

the difference of two squares identity,

$\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$,

can be used to rationalise the $a + b i$ expression.

to rationalise the denominator, multiply it by its conjugate.

the conjugate is found by changing the $-$ sign to $+$.

here, it is $2 + 3 i$.

if both the numerator and denominator are multiplied by $2 + 3 i :$

$4 \cdot \left(2 + 3 i\right) = 8 + 12 i$

$\left(2 + 3 i\right) \left(2 - 3 i\right) = {2}^{2} - {\left(3 i\right)}^{2}$

$= 4 - \left(- 9\right) = 4 + 9$

$= 13$

therefore, the equivalent fraction to $\frac{4}{2 - 3 i}$ where the denominator is rational, is $\frac{8 + 12 i}{13}$.