If $$a_1, a_2, ......., a_8$$ are the roots of the equation $$x^8 + x^7 + ..... + x + 1 = 0$$, them the value of $$a^{2025}_1 + a^{2025}_2 + .... + a^{2025}_8$$ is

$$x^8 + x^7 + ..... + x + 1 = 0$$

Since $$1,x,x^2,x^3,......$$ are in GP, thus we will apply the formula of sum of GP.

=> $$1\left[\dfrac{x^9-1}{x-1}\right]=0$$

=> $$x^9-1=0$$

=> $$x^9=1$$

Now, $$a_1, a_2, ......., a_8$$ are the roots of the equation thus - 

$$\left(a_1\right)^9=\left(a_2\right)^9=\left(a_3\right)^9=....=\left(a_8\right)^9=1$$

We need to find the value of $$a^{2025}_1 + a^{2025}_2 + .... + a^{2025}_8$$.

$$\left(a_1^9\right)^{225}+\left(a_2^9\right)^{225}+....+\left(a_8^9\right)^{225}$$

$$\left(1\right)^{225}+\left(1\right)^{225}+....+\left(1\right)^{225}=1+1+.....+1=8$$