x1 and x2 are solutions of x^2-x+1=0. What is the value of the sum (x1^2-x1)^1000 + (x2^2-x2)^1000 ?

We know the fact that the solution of an equation,
substituted into equation, it cancels out the
equation.

We'll substitute the solutions of the equation
into the given equation:

x1^2 - x1 + 1 =
0

x1^2 - x1 = -1We'll raise to the power of 1000 both
sides:

(x1^2-x1)^1000 =
(-1)^1000

(x1^2-x1)^1000 = 1
(1)

x2^2 - x2 + 1 = 0

x2^2 -
x2 = -1We'll raise to the power of 1000 both
sides:

(x2^2-x2)^1000 =
(-1)^1000

(x2^2-x2)^1000 = 1
(2)

We'll add (1) and
(2):

(x1^2-x1)^1000 + (x2^2-x2)^1000 = 1 +
1

(x1^2-x1)^1000 + (x2^2-x2)^1000 =
2

The value of the sum =
2.