Example 2 Given $2 n$ positive numbers $a_{1}, a_{2}, \cdots, a_{n}, b_{1}$,
$$
\begin{array}{l} 
b_{2}, \cdots b_{n} \text { and } \\
\quad\left(a_{1}+b_{1}\right)\left(a_{2}+b_{2}\right) \cdots\left(a_{n}+b_{n}\right)=10 .
\end{array}
$$

Find the maximum value of $y=\sqrt[n]{a_{1} a_{2} \cdots a_{n}}+\sqrt[n]{b_{1} b_{2} \cdots b_{n}}$.
From Corollary 2, we know
$$
\begin{array}{l}
\sqrt[n]{a_{1} a_{2} \cdots a_{n}}+\sqrt[n]{b_{1} b_{2} \cdots b_{n}} \\
\leqslant \sqrt[n]{\left(a_{1}+b_{1}\right)\left(a_{2}+b_{2}\right) \cdots\left(a_{n}+b_{n}\right)} \\
=\sqrt[n]{10} .
\end{array}
$$

Equality holds, and $y$ attains its maximum value of $\sqrt[n]{10}$, if and only if $\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\cdots=\frac{a_{n}}{b_{n}}$.