Identifying Irrational Numbers Among Square Roots

To determine which number is not irrational, we need to check if the square root results in a rational number. A rational number can be expressed as a simple fraction (like an integer or a terminating/repeating decimal), while an irrational number cannot.

Specifically, the square root of a number is rational if and only if that number is a perfect square. Otherwise, it is irrational.

Analyzing Square Root Options

• Option 1: \(\sqrt{161}\)

  161 is not a perfect square (since \(12^2 = 144\) and \(13^2 = 169\)). Therefore, \(\sqrt{161}\) is an irrational number.

• Option 2: \(\sqrt{63}\)

  63 is not a perfect square (since \(7^2 = 49\) and \(8^2 = 64\)). Therefore, \(\sqrt{63}\) is an irrational number.

• Option 3: \(\sqrt{81}\)

  81 is a perfect square, as \(9^2 = 81\). Thus, \(\sqrt{81} = 9\). The number 9 is an integer, which is a rational number.

• Option 4: \(\sqrt{72}\)

  72 is not a perfect square (since \(8^2 = 64\) and \(9^2 = 81\)). Therefore, \(\sqrt{72}\) is an irrational number.

Conclusion on Number Types

Based on the analysis:

• \(\sqrt{161}\) is irrational.
• \(\sqrt{63}\) is irrational.
• \(\sqrt{81}\) is rational.
• \(\sqrt{72}\) is irrational.

The number that is not an irrational number is \(\sqrt{81}\).