If #alpha=(1/2)(-1+sqrt(-3)) and beta=(1/2)(-1-sqrt(-3))# then prove #alpha^4+(alphabeta)^2+beta^4=0#?

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Answer 1 · 2018-08-12T17:42:59

Please see below.

Here ,

#alpha=1/2(-1+sqrt(-3)) and beta=1/2(-1-sqrt(-3)) #

#:.alpha+beta=1/2{-1+sqrt(-3)-1-sqrt(-3)}=1/2(-2)=-1#

#alpha*beta=1/4{(-1)^2-(sqrt(-3))^2}=1/4{1-(-3)}#=#1/4{4}=1#

#i.e. color(red)(alpha+beta=-1 and alpha*beta=1to(1)#

We know that ,

#(alpha+beta)^2=alpha^2+beta^2+2alphabeta#

#:.(-1)^2=alpha^2+beta^2+2(1)toFrom(1)#

#:.color(blue)(alpha^2+beta^2=1-2=-1to(2)#

We also know that ,

#(alpha^2+beta^2)^2=alpha^4+beta^4+2alpha^2beta^2#

#:.(-1)^2=alpha^4+beta^4+(alphabeta)^2+(alphabeta)^2toFrom (2)#

#:.alpha^4+beta^4+(alphabeta)^2=(alphabeta)^2-1#

#:.alpha^4+beta^4+(alphabeta)^2=(1)^2-1=0#