What numbers are squares and cubes of other numbers?

Answer 1

All non-negative numbers (including zero) are real-number squares of other numbers.

All numbers (positive, negative, and zero) are cubes of other numbers.

"Numbers" generally refers to the set of all real numbers. 2 is the square of a real number, √2. √2 is not a whole number. It's not even rational. But it is a real number, and therefore meets the criteria specified. Even pi (π), an irrational number itself, is the square of another real number (√π). The only real numbers that cannot be squares are negative numbers, because squares of negative numbers are always positive, because the product of two negative numbers is always positive. This restriction, however, does not apply to cubes, because if you multiply three negative numbers by each other, the result is negative.

Now, for the question you may have meant to ask: "What numbers are squares and cubes of whole numbers (integers)?"

It would be impossible to list all squares and cubes of integers, because even when you limit it to integers, there is still an infinite number of numbers, and every one of them has both a square and a cube.

Examples of integers, their squares, and their cubes.

0, 0, 0

1, 1, 1

(-1, 1, -1)

2, 4, 8,

(-2, 4, -8)

3, 9, 27

4, 16, 64

5, 25, 125

10, 100, 1000

Only a very few numbers, such as 0, 1, and 64, are both squares and cubes of integers. Note that 64 is 43 and also 82. (see the related question)