Question

Classify the following numbers as rational or irrational. give reasons to support your answer.
(i) $\sqrt{\frac{3}{81}}$

(ii) $\sqrt{361}$

(iii) $\sqrt{21}$

(iv) $\sqrt{1.44}$

(v) $\frac{2}{3}\sqrt{6}$

(vi) 4.1276

(vii) $\frac{22}{7}$

(viii) 1.232332333...

(ix) 3.040040004....

(x) 2.356565656...

(xi) 6.834834...

Answer 1

(i) $\sqrt{\frac{3}{81}}$
$\sqrt{\frac{3}{81}}=\sqrt{\frac{1}{27}}=\frac{1}{3}\sqrt{\frac{1}{3}}$
It is an irrational number.

(ii) $\sqrt{361}$ = 19
So, it is rational.

(iii) $\sqrt{21}$
$$\begin{gathered} \sqrt{21}=\sqrt{3}\times \sqrt{7}=4.58257... \\ \mathrm{It}\mathrm{is}\mathrm{an}\mathrm{irrational}\mathrm{number}. \end{gathered}$$

(iv) $\sqrt{1.44}$ = 1.2
So, it is rational.

(v) $\frac{2}{3}\sqrt{6}$
It is an irrational number

(vi) 4.1276
It is a terminating decimal. Hence, it is rational.

(vii) $\frac{22}{7}$
$\frac{22}{7}\mathrm{is}\mathrm{a}\mathrm{rational}\mathrm{number}\mathrm{because}\mathrm{it}\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{in}\mathrm{the}\frac{p}{q}\mathrm{form}.$

(viii) 1.232332333...$\mathrm{is}\mathrm{an}\mathrm{irrational}\mathrm{number}\mathrm{because}\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{non}-\mathrm{terminating},\mathrm{non}-\mathrm{repeating}\mathrm{decimal}.$

$$\begin{gathered} \left(\mathrm{ix}\right)3.040040004...\mathrm{is}\mathrm{an}\mathrm{irrational}\mathrm{number}\mathrm{because}\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{non}-\mathrm{terminating},\mathrm{non}-\mathrm{repeating}\mathrm{decimal}. \\ \left(\mathrm{x}\right)2.356565656...\mathrm{is}\mathrm{a}\mathrm{rational}\mathrm{number}\mathrm{because}\mathrm{it}\mathrm{is}\mathrm{repeating}. \end{gathered}$$

(xi) 6.834834... is a rational number because it is repeating.