Use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi

Question

(1−3i​)^5

Answer 1

z = 1 - 3i

||z|| = √10

θ = -71.565º

z⁵ = (√10)⁵(cos (-357.825) + i sin (-357.825))

z⁵ = 100√10(cos 2.175º + i sin 2.175º)

z⁵ = 100√10(0.999 + 0.038i)

z⁵ = 99.9√10 + 3.8i√10

z⁵ = 315.91 + 12.01i

Multiplied out by hand (1 - 3i)⁵ = 316 + 12i (Check it out)

Answer 2

z = r (cos T + i sin T)

r = sqrt( a^2 + b^2)

T = arctan(b/a)

a = r cos T

b = r sin T

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1 - 3i = 1 + -3i = a + bi

so a = 1 and b = -3

a = r cost T

1 = r cost T

b = r sin T

-3 = r sin T

Then

-3/1 = r sin T/ r cos T

-3 = tan t

t = arcsin(-3)

r = sqrt( a^2 + b^2) = sqrt(10)

The polar form is sqrt(10) * ( cos(arcsin(-3)) + sin(arcsin(-3))*i )

DeMoivre says:

100*sqrt(10) ( 5 * cos(arcsin(-3)) + 5*sin(arcsin(-3)*i)