Question

what are squares of a rational number?
9
-9
0.9
0.81
0.0081

Answer 1

A rational number q is the square of a rational number exactly when q ≥ 0 and, written in lowest terms p/r, both p and r are perfect squares.

Applying that:

- 9: Yes. 9 = 3^2 (rational).
- -9: No. Negative numbers are not squares of real rationals.
- 0.9 = 9/10: No. 9 is a square but 10 is not, so √(9/10) is irrational.
- 0.81 = 81/100: Yes. 81 = 9^2 and 100 = 10^2, so 0.81 = (9/10)^2.
- 0.0081 = 81/10000: Yes. 81 = 9^2 and 10000 = 100^2, so 0.0081 = (9/100)^2.