Q.3 (b) Solve x⁶+1=0 .

x⁶=-1=cosπ+isinπ=cos(2k+1)π+isin(2k+1)π . By De Moivre: x=cos(2k+1)π/6+isin(2k+1)π/6,\ k=0,1,dots,5 . k=0:\ fracsqrt32+i/2 ; k=1:\ i ; k=2:\ -fracsqrt32+i/2 ; k=3:\ -fracsqrt32-i/2 ; k=4:\ -i ; k=5:\ fracsqrt32-i/2 . Roots: ± i,\ ±sqrt3/2±i/2 (all four sign combinations). (Check: i⁶=-1 .)

Solve \(x^6+1=0\).

Q.3 (b) Solve \(x^6+1=0\).

Doubtly Asked 25 Oct 2026 mu engineering Save WhatsApp Report

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\(x^6=-1=\cos\pi+i\sin\pi=\cos(2k+1)\pi+i\sin(2k+1)\pi\).

By De Moivre: \(x=\cos\frac{(2k+1)\pi}{6}+i\sin\frac{(2k+1)\pi}{6},\ k=0,1,\dots,5\).

\(k=0:\ \frac{\sqrt3}2+\frac i2\); \(k=1:\ i\); \(k=2:\ -\frac{\sqrt3}2+\frac i2\); \(k=3:\ -\frac{\sqrt3}2-\frac i2\); \(k=4:\ -i\); \(k=5:\ \frac{\sqrt3}2-\frac i2\).

Roots: \(\pm i,\ \pm\frac{\sqrt3}{2}\pm\frac i2\) (all four sign combinations). (Check: \(i^6=-1\).)

Doubtly AI · 25 Oct 2026 AI-verified · Oct 2026 Report

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Solve \(x^6+1=0\).

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