1. The number of correct statements among the following is ( )
(1) All infinite decimals are irrational numbers; (2) If the square root of $A$ is a rational number, then the cube root of $A$ must be an irrational number;
(3) For an irrational number $a$, there must exist a positive integer $n$ such that $a^{n}$ is a rational number;
(4) The reciprocal and the opposite of an irrational number are both irrational numbers.
(A) 0
(B) 1
(C) 2
(D) 3
Answer: (B)
Solution: Infinite decimals include repeating and non-repeating infinite decimals, where repeating infinite decimals all belong to rational numbers, so (1) is incorrect; there are many numbers whose square roots and cube roots are rational numbers, such as: $0, 1, 64, \ldots$, so (2) is incorrect;
$\pi$ is an irrational number, it cannot be expressed in the form of $\sqrt[n]{a}$ (where $a$ is a rational number, $n$ is a positive integer), therefore, there does not exist a positive integer $n$ such that the $n$-th power of $\pi$ is a rational number, so (3) is incorrect;

Only (4) is correct. There is only one correct statement.
Therefore, the answer is (B).