How do you simplify (5-i)/(3+3i)?

Nov 30, 2015

$\frac{2}{3} - i$

Explanation:

You must use something that resembles rationalization with roots at the denominator: multiply both numerator and denominator by $3 - 3 i$:

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

and use the fact that

$\left(3 + 3 i\right) \left(3 - 3 i\right) = {3}^{2} - {\left(3 i\right)}^{2} = 9 - 9 {i}^{2} = 9 + 9 = 18$

Then, expand the numerator as usual:

$\left(5 - i\right) \left(3 - 3 i\right)$

$= 15 - 15 i - 3 i + 3 {i}^{2}$

$= 15 - 18 i - 3$

$= 12 - 18 i$

We can simplify something:

$\frac{12 - 18 i}{18} = \frac{2}{3} - i$