How prove this limits $\lim_{n\to\infty}\frac{v_{5}(1^1\cdot 2^2\cdot 3^3\cdot 4^4\cdots\cdot n^n)}{n^2}=\frac{1}{8}$

Group Theory via Category Theory

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Can someone prove why $\sqrt{ab}=\sqrt{a}\sqrt{b}$ is only valid when a and b are positive?

$\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$

Evaluate $\int \frac{\mathrm dx}{1+\cos^2 x}$

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How many natural numbers less than 200 will have 12 factors/divisors?

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