How to generalize it?

Let the product be $P$ , then we have:

$\begin{aligned} P & = \left \lfloor \sqrt{2} \right \rfloor \times \left \lfloor \sqrt{3} \right \rfloor \times \left \lfloor \sqrt{\color{#3D99F6}{4}} \right \rfloor \times \left \lfloor \sqrt{5} \right \rfloor \times \left \lfloor \sqrt{6} \right \rfloor \times \left \lfloor \sqrt{7} \right \rfloor \times \left \lfloor \sqrt{8} \right \rfloor \times \left \lfloor \sqrt{\color{#D61F06}{9}} \right \rfloor \times ... \times \left \lfloor \sqrt{99} \right \rfloor \\ & = 1 \times 1 \times \color{#3D99F6}{2} \times 2 \times 2 \times 2 \times 2 \times \color{#D61F06}{3} \times ... \times 9 \quad \quad \small \color{#3D99F6}{\text{Note that }\left \lfloor \sqrt{m} \right \rfloor \text{ increases by 1 when m is a perfect square.}} \\ & = 1^{4-2}2^{9-4}3^{16-9}...k^{(k+1)^2-k^2}...9^{100-81} \\ & = \prod_{k=2}^9 k^{2k+1} \\ & = \color{#3D99F6}{2^5}3^7\color{#D61F06}{4^9}5^{11}\color{#3D99F6}{6^{13}}7^{15} \color{#D61F06}{8^{17}}9^{19} \\ & = \color{#3D99F6}{2^5} \color{#D61F06}{(2^2)^9} \color{#3D99F6}{(2^{13}3^{13})}\color{#D61F06}{(2^3)^{17}} 3^7 5^{11}7^{15} 9^{19} \\ & = 2^{\color{#3D99F6}{5}+ \color{#D61F06}{18}+ \color{#3D99F6}{13} + \color{#D61F06}{51}} 3^{20} 5^{11}7^{15} 9^{19} \\ & = 2^{\color{#3D99F6}{87}} 3^{20} 5^{11}7^{15} 9^{19} \end{aligned}$

$\Rightarrow n = \boxed{\color{#3D99F6}{87}}$ .

The multiplication simplifies to: $1^{2}\cdot 2^{5}\cdot 3^{7}\cdot 4^{9}\cdot 5^{11}\cdot 6^{13}\cdot 7^{15}\cdot 8^{17}\cdot 9^{19}$ Or: $2^{5}\cdot 2^{18}\cdot 2^{13}\cdot 2^{51}\cdot k$ where $k$ is odd. So, $n=5+18+13+51=87$ .

n=1:- 4~~to~~8. .......1 * 5 ...............= 5
n=2:-16~~to~~24......2 * 9 ...............=18
n=1:-36~~t0~~48......1 * 13 ..............=13
n=3:-64~~to~~80......3 * 17 ..............=51
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~~~~~~~~~~~~~~~~~~~~~~~~~~~= $\Large \color{#D61F06}{87}.$