A Long Equation II

$\begin{aligned} P & = \sqrt 2 \times \sqrt 6 \times \sqrt{12} \times \sqrt{20} \times \cdots \times \sqrt{72} \\ & = \sqrt {2 \times 6 \times 12 \times 20 \times \cdots 72} \\ & = \sqrt {(1 \times 2) \times (2\times 3) \times (3 \times 4) \times (4 \times 5) \times \cdots \times (8 \times 9)} \\ & = \sqrt {1 \times 2^2 \times 3^2 \times 4^2 \times 5^2 \times \cdots \times 8^2 \times 9} \\ & = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 3 \\ & = \boxed{120960} \end{aligned}$

It’s easy to see the pattern $\sqrt{n\times (n+1)}$ , therefore we use it to search the regulations. $\begin{aligned} P &=\sqrt{2}\times \sqrt{6}\times \sqrt{12}\times \sqrt{20}\times \cdots \times \sqrt{72} \\ &=\sqrt{1\times (1+1)\times 2\times (2+1)\times 3\times (3+1)\times \cdots \times 8 \times (8+1)} \\ &=\sqrt{1\times 2\times 2\times 3\times 3\times 4\times \cdots \times 8\times 9} \\ &=1\times 2\times 3\times 4\times \cdots \times 8\times 3 \\ &=120960 \end{aligned}$