Compute (2 cis(π/4))^4.

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Multiple Choice

Compute (2 cis(π/4))^4.

Explanation:
Raising a complex number in polar form to a power uses De Moivre's theorem: (r cis θ)^n = r^n cis(nθ). Here r = 2, θ = π/4, and n = 4. The magnitude becomes 2^4 = 16 and the angle becomes 4 × (π/4) = π. So the result is 16 cis π, which equals 16(cos π + i sin π) = 16(-1) = -16. An independent check in rectangular form gives the same: 2 cis π/4 = √2 + i√2, and (√2 + i√2)^4 = -16.

Raising a complex number in polar form to a power uses De Moivre's theorem: (r cis θ)^n = r^n cis(nθ). Here r = 2, θ = π/4, and n = 4. The magnitude becomes 2^4 = 16 and the angle becomes 4 × (π/4) = π. So the result is 16 cis π, which equals 16(cos π + i sin π) = 16(-1) = -16. An independent check in rectangular form gives the same: 2 cis π/4 = √2 + i√2, and (√2 + i√2)^4 = -16.

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