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So given

x² + x + 1 = 0 ⇒ x² + x + 1 - 1 + 1 = 0 ⇒ x² + x - 1 = -2

so

so x(x² + x + 1) = x³ + x² + x + 1 - 1

x² + x + 1 = (x³ + 1)/x + (x² + x - 1)/x

0 = (x³ + 1)/x + (x² + x - 1)/x

- (x² + x - 1)/x = (x³ + 1)/x

- ( - 2) = (x³ + 1)

x³ + 1 = 2

so value for

x² + x + 1 = 0 ⇒ x² + x + 1 - 1 + 1 = 0 ⇒ x² + x - 1 = -2

so

so x(x² + x + 1) = x³ + x² + x + 1 - 1

x² + x + 1 = (x³ + 1)/x + (x² + x - 1)/x

0 = (x³ + 1)/x + (x² + x - 1)/x

- (x² + x - 1)/x = (x³ + 1)/x

- ( - 2) = (x³ + 1)

x³ + 1 = 2

so value for

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as x^2+x+1=0 it is obvious that x is not equal to 1.

x^2+x+1=0

(x-1)(x^2+x+1)=0

x^3-1=0

x^3=1

so x is the cube root of unity,

x = 1, w, w^2... here w is omega

now x^3+1 = w^3 + 1 = 1 + 1 = 2

and 1/w = w^2

So, ( (x^3+1)/x )^3 = (2 w^2)^3 = 8 w^6 = 8 (w^3)^2 = 8

this is the correct way of solving...