If the square root is a whole number, then the square of the square root, the original number, is also a whole number; all whole numbers can be expressed as themselves over 1, and so are rational numbers.
The answer is thus any square number, ie the square of the natural numbers: 1 (1²), 4 (2²), 9 (3²), 16 (4²), etc.
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It is another rational number whose numerator and denominator (in the ratio's simplest form) are perfect squares.
-27 is one possible answer.
27 is a perfect cube.
A perfect square.
That's a perfect cube.