Connor Randall

2022-02-02

How do you find the power ${\left(3-5i\right)}^{4}$ and express the result in rectangular form?

eris0cg

Expert

Step 1
We can use the binomial theorem to expand the expression. (The binomial coefficient from Pascals triangle are 1, 4, 6, 4, 1)
${\left(3-5i\right)}^{4}={\left(3\right)}^{4}+4{\left(3\right)}^{3}\left(-5i\right)+6{\left(3\right)}^{2}{\left(-5i\right)}^{2}+4\left(3\right){\left(-5i\right)}^{3}+{\left(-5i\right)}^{4}$
$=81+108\left(-5i\right)+54\left(25{i}^{2}\right)+12\left(-125{i}^{3}\right)+\left(625{i}^{4}\right)$
$=81-540i+1350\left(-1\right)-1500\left(-i\right)+625\left(1\right)$
$=81-540i-1350+1500i+625$
$=-644+960i$

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