Question

Which one of the following is an irrational number:

• A. $\surd 4$
• B. $3\surd 8$
• C. $\surd 100$
• D. $-\surd 0.64$

Answer 1

The correct option is B

$3\surd 8$

Determine an Irrational Number.

The explanation for the correct option:

Option (B): $3\surd 8$ The irrational number is one that cannot be stated as the quotient of two integers i.e. $\frac{p}{q}$ where $p\mathit{}\mathrm{and}\mathit{}q$ are integr $q\ne 0$ and it can be expressed as a non-terminating, non-recurring decimal

$$\begin{gathered} 3\sqrt{8}=3\sqrt{2\times 2\times 2} \\ \Rightarrow 3\sqrt{8}=3\times 2\sqrt{2} \\ \Rightarrow 3\sqrt{8}=6\sqrt{2} \end{gathered}$$

Since it can't be expressed in form of $\frac{p}{q}$,where $p\mathit{}\mathrm{and}\mathit{}q$ are integer $q\ne 0$.

Hence, it is an irrational number

The explanation for the incorrect options:

Option (A): $\surd 4$

$$\begin{gathered} \sqrt{4}=\sqrt{2\times 2} \\ \Rightarrow \sqrt{4}=2=\frac{2}{1} \end{gathered}$$

Since, it can be expressed in form of $\frac{p}{q}$,where $p\mathit{}\mathrm{and}\mathit{}q$ are integr $q\ne 0$.

Hence, It is a rational number.

Option (C): $\surd 100$

$$\begin{gathered} \sqrt{100}=\sqrt{10\times 10} \\ \Rightarrow \sqrt{100}=10=\frac{10}{1} \end{gathered}$$

Since, it can be expressed in form of $\frac{p}{q}$,where $p\mathit{}\mathrm{and}\mathit{}q$ are integr $q\ne 0$.

Hence, It is a rational number.

Option (D): $-\surd 0.64$

$$\begin{gathered} -\sqrt{0.64}=-\sqrt{\frac{64}{100}} \\ \Rightarrow -\sqrt{0.64}=-\sqrt{\frac{8\times 8}{10\times 10}} \\ \Rightarrow -\sqrt{0.64}=-\frac{8}{10} \\ \Rightarrow -\sqrt{0.64}=-0.8 \end{gathered}$$

Since, it can be expressed in form of $\frac{p}{q}$,where $p\mathit{}\mathrm{and}\mathit{}q$ are integer. $q\ne 0$

Hence,It is a rational number.

Hence, “Option B, Option A, and Option D” is incorrect options.

Hence, the correct choice is “Option B”.