To determine which of the given square roots is a rational number, we need to consider the definitions and properties of rational numbers and perfect squares.
A rational number is any number that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]. Perfect squares, on the other hand, are numbers that can be expressed as the square of an integer.
We will evaluate the square roots of the given numbers to see which one(s) result in an integer (and therefore a rational number):
1. [tex]\(\sqrt{32}\)[/tex]:
[tex]\[ \sqrt{32} \approx 5.656854249492381 \][/tex]
This is not an integer.
2. [tex]\(\sqrt{81}\)[/tex]:
[tex]\[ \sqrt{81} = 9.0 \][/tex]
This is an integer, specifically [tex]\(9\)[/tex], which is a rational number since [tex]\(9\)[/tex] can be expressed as [tex]\(\frac{9}{1}\)[/tex].
3. [tex]\(\sqrt{90}\)[/tex]:
[tex]\[ \sqrt{90} \approx 9.486832980505138 \][/tex]
This is not an integer.
4. [tex]\(\sqrt{125}\)[/tex]:
[tex]\[ \sqrt{125} \approx 11.180339887498949 \][/tex]
This is not an integer.
Among the given numbers [tex]\(\sqrt{32}\)[/tex], [tex]\(\sqrt{81}\)[/tex], [tex]\(\sqrt{90}\)[/tex], and [tex]\(\sqrt{125}\)[/tex], only [tex]\(\sqrt{81}\)[/tex] results in an integer (9). Therefore, the rational number among the given options is:
[tex]\[ \boxed{\sqrt{81}} \][/tex]
