Example 2 Find the standard prime factorization of 20!.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
The prime numbers not exceeding 20 are $2,3,5,7,11,13,17,19$. By Theorem 2, we have
$$\begin{aligned}
\alpha(2,20) & =\left[\frac{20}{2}\right]+\left[\frac{20}{4}\right]+\left[\frac{20}{8}\right]+\left[\frac{20}{16}\right] \\
& =10+5+2+1=18 \\
\alpha(3,20) & =\left[\frac{20}{3}\right]+\left[\frac{20}{9}\right]=6+2=8 \\
\alpha(5,20) & =\left[\frac{20}{5}\right]=4 . \quad \alpha(7,20)=\left[\frac{20}{7}\right]=2 \\
\alpha(11,20) & =\left[\frac{20}{11}\right]=1 . \quad \alpha(13,20)=\left[\frac{20}{13}\right]=1 \\
\alpha(17,20) & =\left[\frac{20}{17}\right]=1 . \quad \alpha(19,20)=\left[\frac{20}{19}\right]=1
\end{aligned}$$

Therefore,
$$20!=2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19$$