Use demoivre's theorem to evaluate the expression [sqrt 3( cos 5pi/3 + i sin 5pi/3)]^4 ? write the answer in rectangular form a. 9sqrt3/2 + 9/2 i b. 9sqrt3/2 - 9/2 i c. -9/2 - 9sqrt3/2 i d. -9/2 + 9sqrt3/2 i

DeMoivre's theorem

if z = a ( cos θ + i sin θ)
∴ z^n = a^n ( cos nθ + i sin nθ)

For the given complex number ⇒⇒⇒ [ √3 ( cos 5π/3 + i sin 5π/3 ) ]⁴
[ √3 ( cos 5π/3 + i sin 5π/3 ) ]⁴ = (√3)⁴  ( cos 4*5π/3 + i sin 4*5π/3 )
= 9 ( cos 20π/3 + i sin 20π/3 )
= 9 ( cos 2π/3 + i sin 2π/3 )
 = 9 ( -1/2 + i √3 /2 )
= -9/2 + 9√3 /2 i

note: 20π/3 = 2π/3 + 6π = 2π/3 + 3 *2π = 2π/3

∴ The correct answer is option d
   d. -9/2 + 9sqrt3/2 i

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