Exponential product

For a fixed $n$ , we see that the expression inside is $\left(\frac{2000}{2}\right)^{1/n^2}\left(\frac{2000}{2^2}\right)^{1/n^2}\left(\frac{2000}{2^3}\right)^{1/n^2}\left(\frac{2000}{2^4}\right)^{1/n^2}=\left(\frac{2000^4}{2^{10}}\right)^{1/n^2}.$ It is not hard to see that $\frac{2000^4}{2^{10}}=50^6$ . Therefore, $\begin{aligned} P &= \prod_{n=1}^{\infty} (50^6)^{/n^2}\\ &= (50^6)^{1/1^2+1/2^2+1/3^2+\cdots}\\[5pt] &= (50^6)^{\pi^2/6}\\[5pt] &= 50^{\pi^2}. \end{aligned}$ Note that $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots=\frac{\pi^2}{6}$ is a well-known telescopic summation. Then we have $A=50$ and $B=2,$ so $\dfrac{A}{B}=25$ .