# How do you find the power (3-5i)^4 and express the result in rectangular form?

Mar 23, 2017

${\left(3 - 5 i\right)}^{4} = - 644 + 960 i$

#### Explanation:

We can use the binomial theorem to expand the expression. (The binomial coefficient from Pascals triangle are $1 , 4 , 6 , 4 , 1$)

${\left(3 - 5 i\right)}^{4} = {\left(3\right)}^{4} + 4 {\left(3\right)}^{3} \left(- 5 i\right) + 6 {\left(3\right)}^{2} {\left(- 5 i\right)}^{2} + 4 \left(3\right) {\left(- 5 i\right)}^{3} + {\left(- 5 i\right)}^{4}$
$\text{ } = 81 + 108 \left(- 5 i\right) + 54 \left(25 {i}^{2}\right) + 12 \left(- 125 {i}^{3}\right) + \left(625 {i}^{4}\right)$
$\text{ } = 81 - 540 i + 1350 \left(- 1\right) - 1500 \left(- i\right) + 625 \left(1\right)$
$\text{ } = 81 - 540 i - 1350 + 1500 i + 625$
$\text{ } = - 644 + 960 i$