Example 1. Which of the following expressions are meaningful? Among the meaningful radicals, which are arithmetic roots? Express the radicals that are not arithmetic roots using arithmetic roots;
(1) $\sqrt[3]{\frac{1}{8}}$,
(2) $\sqrt{-9}$
(3) $\sqrt[3]{-8}$
Solution: (1) is an arithmetic root;
(2) is meaningless;
(3) is meaningful, but not an arithmetic root,
$$
\sqrt[3]{-8}=-\sqrt[3]{8}=-2 .
$$