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

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Answer 1 · 2018-07-09T10:00:02

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

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