The question asks to prove that the sum of the 99th powers of all the roots of $x^7 - 1 = 0$ is zero. Let's break it down:

Step-by-Step Solution:

1. Roots of the Equation: The equation $x^7 - 1 = 0$ can be rewritten as $x^7 = 1$. The solutions to this are the 7th roots of unity. These roots are given by:

   $\omega_k = e^{2\pi ik/7}, \text{ for } k = 0, 1, 2, ..., 6$ where $\omega_0 = 1$, and the other roots are complex numbers on the unit circle in the complex plane.

2. General Form of the Roots: The 7th roots of unity are: $\omega^0 = 1, \quad \omega^1 = e^{2\pi i/7}, \quad \omega^2 = e^{4\pi i/7}, \quad \dots, \quad \omega^6 = e^{12\pi i/7}$ These are distinct and symmetrically placed on the unit circle.

3. 99th Power of the Roots: We are asked to find the sum of the 99th powers of these roots. For each $k$, we need to compute $\omega_k^{99}$.

   Since $\omega_k^7 = 1$, we can reduce the exponent $99$ modulo $7$: $99 \mod 7 = 1$ Thus, $\omega_k^{99} = \omega_k^1 = \omega_k$.

4. Summing the 99th Powers: Therefore, the sum of the 99th powers of the roots is equivalent to the sum of the roots themselves: $\sum_{k=0}^{6} \omega_k^{99} = \sum_{k=0}^{6} \omega_k$

5. Sum of the Roots of Unity: The sum of all 7th roots of unity is well known to be zero. This is because the 7th roots of unity form a regular polygon in the complex plane, and their sum (as vectors) cancels out to zero: $\sum_{k=0}^{6} \omega_k = 0$

Conclusion:

Thus, the sum of the 99th powers of all the roots of $x^7 - 1 = 0$ is:

$\boxed{0}$

Let me know if you would like more details or clarifications!

Here are 5 related questions:

1. What is the general formula for the sum of the nth roots of unity?
2. How do the powers of complex numbers behave on the unit circle?
3. Can the sum of roots of any polynomial equation always be zero?
4. How are roots of unity used in Fourier transforms or signal processing?
5. What happens if we change the exponent from 99 to another value?

Tip: Roots of unity are periodic, meaning powers of these roots repeat in cycles, which simplifies exponentiation significantly when dealing with large powers.