If a+1/a=-1 then a^2016+1/(a^2016)=???

Question

Help me with a step-by-step solution please.

Answer 1

a +1/a =-1

a^2 +1 =-a

a^2 +a + 1 = 0

a^2 +a + 1/4 =-1 +1/4 =-3/4

(a+1/2)^2 = -3/4

a+1/2 = sqr(-3/4) = isqr(3)/4

a = -1/2 + or - (i/4)sqr3

replace a with a^2016 to get

a^2016 = -1/2 or -(i/4)sqr3

a = the 2016th root of (-1/2 + or - (i/4)sqr3)

but maybe you really meant (a+1)/a =-1

then

a+1 =-a

2a =-1

a =-1/2

replace a with a^2016 to get

a^2016 =-1/2

a = the 2016th root of -1/2