Math

use Demoivre's Theorem to find the indicated power of the complex number. Express the result in standard form.

(2+2i)^6

a=2 b=2 n=6

r=sqrt 2^2 + 2^2 = sqrt8
Q=7pi/4

(sqrt8)^6 = 512
512(cos 6/1 x 7pi/4) + i sin 6 x 7pi/4
512 (cos 21pi/2) + i sin (21pi/2)
512(cos pi/2) + i (sin pi/2)
512(2+2i)

1024 + 1024i

Is this correct?

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  1. i=253

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  2. Rewrite (2+2i)^6 as

    [2^3/2*(cos pi/4 + i sin pi/4)]^6
    = 2^9*[cos 6pi/4 * i sin 6pi/4]
    = 512* i sinpi/2
    = -512 i

    I see several error in your derivation.

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  3. Where did you get the angle of 7π/4 ?

    I get an angle of π/4

    check:
    √8(cos π/4 + i sin π/4_
    = √8(1/√2 + i (1/√2)
    = 2 + 2i

    so by De Moivre's Theorem
    (2+2i)^6
    = √8^6(cos (6π/4) + i sin (6π/4))
    = 512( + i(-1))
    = - 512i

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