Question

determine whether each number is rational or irrational.

	rational	irrational
$sqrt{25}$	$circ$	$circ$
$sqrt{144}$	$circ$	$circ$
$sqrt{66}$	$circ$	$circ$

Explanation:

Step1: Recall the definition of rational and irrational numbers

A rational number is a number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q
eq0\). For square roots, if \(\sqrt{n}\) is an integer (i.e., \(n\) is a perfect square), then \(\sqrt{n}\) is rational; otherwise, if \(n\) is not a perfect square, \(\sqrt{n}\) is irrational (since it cannot be expressed as a fraction of two integers).

Step2: Analyze \(\sqrt{98}\)

Factorize \(98\): \(98 = 49\times2=7^{2}\times2\). Then \(\sqrt{98}=\sqrt{7^{2}\times2} = 7\sqrt{2}\). Since \(\sqrt{2}\) is irrational, \(7\sqrt{2}\) is irrational. So \(\sqrt{98}\) is irrational.

Step3: Analyze \(\sqrt{25}\)

We know that \(5^{2}=25\), so \(\sqrt{25} = 5\), which is an integer and thus can be written as \(\frac{5}{1}\) (a fraction of two integers). So \(\sqrt{25}\) is rational.

Step4: Analyze \(\sqrt{144}\)

Since \(12^{2}=144\), \(\sqrt{144}=12\), which is an integer. So \(\sqrt{144}\) is rational.

Step5: Analyze \(\sqrt{66}\)

Factorize \(66 = 2\times3\times11\). There is no integer \(k\) such that \(k^{2}=66\) (because \(8^{2}=64\) and \(9^{2}=81\), and 66 is between them and not a perfect square). So \(\sqrt{66}\) is irrational.

Answer:

• \(\boldsymbol{\sqrt{98}}\): Irrational (select the circle under "Irrational")
• \(\boldsymbol{\sqrt{25}}\): Rational (select the circle under "Rational")
• \(\boldsymbol{\sqrt{144}}\): Rational (select the circle under "Rational")
• \(\boldsymbol{\sqrt{66}}\): Irrational (select the circle under "Irrational")