Question

Which of the following is a rational number: 32−−√ , 81−−√ , 90−−√ , or 125−−−√ ?(1 point)
explain

Answer 1

name all the irrartional numbers

Answer 2

A rational number is defined as a number that can be expressed as the quotient or fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). To determine which of the given options is a rational number, we need to evaluate the square roots:

1. \(\sqrt{32}\)
2. \(\sqrt{81}\)
3. \(\sqrt{90}\)
4. \(\sqrt{125}\)

Now, let's analyze each square root:

1. **\(\sqrt{32}\)**: This can be simplified as \(\sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}\). Since \(\sqrt{2}\) is an irrational number, \(\sqrt{32}\) is also irrational.

2. **\(\sqrt{81}\)**: This simplifies to \(\sqrt{9 \cdot 9} = 9\). Since \(9\) is an integer, it is a rational number.

3. **\(\sqrt{90}\)**: This can be expressed as \(\sqrt{9 \cdot 10} = \sqrt{9} \cdot \sqrt{10} = 3\sqrt{10}\). Since \(\sqrt{10}\) is also irrational, \(\sqrt{90}\) is irrational.

4. **\(\sqrt{125}\)**: This simplifies to \(\sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}\). Again, since \(\sqrt{5}\) is irrational, \(\sqrt{125}\) is irrational.

From the analysis above, the only rational number among the options is:

**\(\sqrt{81} = 9\)**.