Question

which number is irrational?
a. $sqrt{4}$
b. $sqrt{35}$
c. $sqrt{49}$
d. $sqrt{64}$

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 \(n\) is a perfect square, \(\sqrt{n}\) is rational; if \(n\) is not a perfect square, \(\sqrt{n}\) is irrational (for positive integers \(n\)).

Step2: Analyze option A

For \(\sqrt{4}\), we know that \(2\times2 = 4\), so \(\sqrt{4}=2\), which is a rational number (it can be written as \(\frac{2}{1}\)).

Step3: Analyze option B

For \(\sqrt{35}\), we check if 35 is a perfect square. The perfect squares around 35 are \(5^2 = 25\) and \(6^2=36\). Since 35 is not a perfect square, \(\sqrt{35}\) has a non - repeating, non - terminating decimal expansion, so it 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 \(\sqrt{64}\), we know that \(8\times8 = 64\), so \(\sqrt{64}=8\), which is a rational number (it can be written as \(\frac{8}{1}\)).

Answer:

B. \(\sqrt{35}\)