When α is a rational number, α=p/q (p and q are integers, and P and q are coprime).

When q is odd, x and y can be positive or negative; When q is an even number, x and y must be nonnegative.

When α is an irrational number, X and Y can only be non-negative, because X α and Y α are determined by the limit of α' s deficiency and residue approximation. This approximation must be a rational number, which can be expressed as p/q (both p and q are integers, and p and q are coprime), but not all defects and residuals are expressed as p/q, and q must be an odd number! This makes the limit nonexistent when x and y are negative numbers.

Therefore, when α is an irrational number, the power function with α as the power exponent stipulates that the base is greater than or equal to 0, and the base is less than 0, which is meaningless.