answer:Using De Moivre's theorem, we can simplify the expression as follows: left(-2 sqrt{3} left(cos left(frac{31pi}{180}right)+i sin left(frac{31pi}{180}right)right)right)^9 = (-2 sqrt{3})^9 left(cos left(frac{31pi}{180}right)+i sin left(frac{31pi}{180}right)right)^9 = (-2 sqrt{3})^9 left(cos left(9 cdot frac{31pi}{180}right)+i sin left(9 cdot frac{31pi}{180}right)right) = (-2 sqrt{3})^9 left(cos left(frac{31pi}{20}right)+i sin left(frac{31pi}{20}right)right) = (-2 sqrt{3})^9 left(cos left(frac{31pi}{10}right)+i sin left(frac{31pi}{10}right)right) = (-2 sqrt{3})^9 left(cos left(frac{31pi}{10}right)+i sin left(frac{31pi}{10}right)right) = -41472 sqrt{3} left(cos left(frac{31pi}{10}right)+i sin left(frac{31pi}{10}right)right) Therefore, the simplified expression is -41472 sqrt{3} left(cos left(frac{31pi}{10}right)+i sin left(frac{31pi}{10}right)right). The answer is -41472 sqrt{3} left(cos left(frac{31pi}{180}right)+i sin left(frac{31pi}{180}right)right)

answer:b. As the number of coin flips increases, the observed proportion of "Heads" will converge with high probability to the expected value, which is 50% for a fair coin. This is a consequence of the Law of Large Numbers, which asserts that the sample mean becomes a better approximation of the true expected value as the sample size grows. It is important to note that: - a) is a correct statement but not a direct consequence of the Law of Large Numbers as stated; it's related to the concept of asymptotic probability. - c) is not entirely accurate, as the sample mean approaches 0.5 as the number of flips increases, not necessarily "always" regardless of the number of flips.

answer:The derivative of f(x) with respect to x is given by: f'(x) = frac{d}{dx}[log (5x)] - frac{d}{dx}[cos(7x^2)] Applying the rules of differentiation: f'(x) = frac{1}{5x} cdot 5 - (-7sin(7x^2)) cdot 2x f'(x) = frac{1}{x} + 14xsin(7x^2) Thus, the derivative of the function is: f'(x) = frac{1}{x} + 14xsin(7x^2)

answer:The alternative argument by Landau and Lifchitz does offer a perspective on the equal probability postulate but is not without its subtleties. Starting with the premise that a probability distribution rho(x;t) governs the microstate of a system, it evolves over time according to Liouville's theorem while remaining constant along phase flow trajectories. An equilibrium distribution is one that ceases evolving, rho(x;t) = rho(x;t+Delta t). From here, Landau's argument suggests that log rho(x) must be a constant of the motion, limiting the number of additive integrals to energy, momentum, and angular momentum. However, the argument's validity relies on the assumption that the probability distribution depends only on these constants, which is not necessarily true for all systems. For instance, in a system with multiple oscillators, the equilibrium distribution might depend on individual energies, rather than just the total energy. The assumption that rho is additive and depends only on the conserved quantities E, P, L may lead to incorrect conclusions, as it neglects possible dependencies on subsystem energies. The ergodic theorem plays a crucial role in understanding this issue. To derive the microcanonical ensemble, one must introduce additional assumptions, like ergodicity, which asserts that all accessible microstates are equally likely in the long run. This is not a direct consequence of classical mechanics and serves as a phenomenological principle supported by experimental observations. In conclusion, while Landau and Lifchitz's argument offers some insight, it does not entirely replace the "a priori equal probability postulate" without introducing new assumptions. The microcanonical ensemble still requires the assumption of equal probability for all accessible microstates with the same energy, though the ergodic theorem provides a more physical foundation for this assumption.