Question

8 which of these numbers is an irrational number? a √81 b √28 c √49 d √36

Explanation:

Step1: Recall irrational number definition

An irrational number is a number that cannot be expressed as a fraction of two integers and has a non - repeating, non - terminating decimal expansion. For square roots, if the number inside the square root (\(radicand\)) is a perfect square, then the square root is a rational number; if the radicand is not a perfect square, the square root is irrational (for positive integers).

Step2: Analyze Option A

For \(\sqrt{81}\), we know that \(9\times9 = 81\), so \(\sqrt{81}=9\), which is a rational number (it can be written as \(\frac{9}{1}\)).

Step3: Analyze Option B

For \(\sqrt{28}\), we factorize \(28 = 4\times7\). Then \(\sqrt{28}=\sqrt{4\times7}=\sqrt{4}\times\sqrt{7}=2\sqrt{7}\). Since \(7\) is not a perfect square, \(\sqrt{7}\) is irrational, and multiplying it by \(2\) still gives an irrational number. So \(\sqrt{28}\) is irrational.

Step4: Analyze Option C

For \(\sqrt{49}\), we know that \(7\times7 = 49\), so \(\sqrt{49}=7\), which is a rational number (it can be written as \(\frac{7}{1}\)).

Step5: Analyze Option D

For \(\sqrt{36}\), we know that \(6\times6 = 36\), so \(\sqrt{36}=6\), which is a rational number (it can be written as \(\frac{6}{1}\)).

Answer:

B. \(\boldsymbol{\sqrt{28}}\)