Example 10 Find the tenth root of 194400.
Since $194400=2^{2} \times 3^{5} \times 5^{2}$, we have
$$\sqrt[10]{194400}=\sqrt{2} \times \sqrt{3} \times \sqrt[5]{5}$$

From Table 1 we have
$$\sqrt{2}=1.41421356 \cdots, \quad \sqrt{3}=1.73205080 \cdots$$

From Table 2 we have
$$\sqrt[5]{5}=1.37972966 \cdots$$

From equations (37) to (39) we have
$$\begin{array}{l}
\sqrt[10]{194400}=(1.41421356 \cdots) \times(1.7320508 \cdots) \\
\times(1.37972966 \cdots)=3.3796336 \cdots
\end{array}$$