Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. (3 − 2i)^6

Answer 1

For what it might be worth here's using Euler's formula, related to DEMoivre's Theorem

the angle is arctan(2/3) = 202.2 degrees=A

r=sqr(9+4) = sqr13. take the power 6 and put it over r or (sqr13)^6

then make 6 a multiple of the power of e^6Ai

(3-2i)^6 = (sqr13)^6 e^6i(202.2) = 13^3 e^180i e^22.2i = -2197(cos22.2 + isin22.2) = -2197(.926+.378i) = -2034 + 830.5i

if you multiply it out by long hand multiplication you get -2035+828i, which is very close. Maybe a couple rounding errors

Answer 2

z = 2 - 3i723

||z|| = √13

θ ≈ -0.982793723^r

z⁶ = (√13)⁶ (cos -5.896762339 + i sin -5.896762339)

2197(0.96263085 + 0,37687756i)

2035 + 828i

QED

Answer 3

3-2i =√(3²+2²)e^tan^-1(-2/3) i

Then, (3-2i)⁶ = √(3²+2²)⁶ e^6 tan^-1(-2/3) i =

13³ e^-3.52801562128541 i =

13³ (cos -3.52801562128541 + i sin -3.52801562128541) =

2197 (-2035/2197 + 828/2197 i) =

-2035 + 828 i