Question

Which of the following is irrational?

• A. $\frac{\sqrt{4}}{\sqrt{9}}$
• B. $\frac{\sqrt{12}}{\sqrt{3}}$
• C. $\sqrt{7}$
• D. $\sqrt{81}$

Answer 1

The correct option is C

$\sqrt{7}$

Explanation for the correct option.

An irrational number is any real number that cannot be expressed as the quotient of two integers that is $\frac{p}{q}$ where, $p$ and $q$ are integers and $q\ne 0$.

Option (C)

$\sqrt{7}=2.64575131106$

It is an irrational number.

Explanation of the incorrect options.

Option (A)

$$\begin{gathered} \frac{\sqrt{4}}{\sqrt{9}}=\frac{\sqrt{2^2}}{\sqrt{3^2}} \\ =\frac{2}{3} \end{gathered}$$

It is the fraction form.

Option (B)

$$\begin{gathered} \frac{\sqrt{12}}{\sqrt{3}}=\frac{\sqrt{2\times 2\times 3}}{\surd 3} \\ =\frac{2\sqrt{3}}{\sqrt{3}} \\ =2 \\ =\frac{2}{1} \end{gathered}$$

It is the fraction form.

Option (D)

$$\begin{gathered} \sqrt{81}=\sqrt{9\times 9} \\ =9 \\ =\frac{9}{1} \end{gathered}$$

It is the fraction form.

Hence, the correct choice is “Option C”.