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

Given: $a_1, a_2, ..., a_8$ are the roots of $x^8 + x^7 + ... + x + 1 = 0$

Find: $a_1^{2025} + a_2^{2025} + ... + a_8^{2025}$

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We have the equation: $x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = 0$

This is a geometric series. Using the formula for sum of a geometric series:

$\frac{x^9 - 1}{x - 1} = 1 + x + x^2 + ... + x^8$

So our equation becomes:

$\frac{x^9 - 1}{x - 1} = 0$

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For a fraction to equal zero, the numerator must be zero and denominator must be non-zero.

Therefore: $x^9 - 1 = 0$ and $x \neq 1$

This gives us: $x^9 = 1$ and $x \neq 1$

So each root $a_1, a_2, ..., a_8$ satisfies: $a_k^9 = 1$

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Now we need to find $a_k^{2025}$ for each root.

Notice that $2025 = 9 \times 225$

Therefore:

$a_k^{2025} = a_k^{9 \times 225} = (a_k^9)^{225} = (1)^{225} = 1$

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Since each of the 8 roots gives us 1 when raised to the power 2025:

$a_1^{2025} + a_2^{2025} + a_3^{2025} + ... + a_8^{2025} = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8$

Answer: $8$