*if*x=1/2(-1+isqrt3)andy=1/2(-1-isqrt3)verify that,x^3+y^(-3)=2?

1 Answer

Changing the given complex number in exponential form as follows

x=1/2(-1+i\sqrt3)=-1/2+i\sqrt3/2=e^{{2\pi}/3i}

y=1/2(-1-i\sqrt3)=-1/2-i\sqrt3/2=e^{-{2\pi}/3i}

Hence,

x^3+y^{-3}

=(e^{{2\pi}/3i})^3+(e^{-{2\pi}/3i})^{-3}

=e^{3\cdot {2\pi}/3i}+e^{3\cdot {2\pi}/3i}

=e^{2\pi i}+e^{2\pi i}

=2e^{2\pi i}

=2(\cos2\pi+i\sin2\pi)

=2(1+0)

=2

Proved