Classify the following numbers as rational or irrational.

Solution:

(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) $\sqrt{\mathbf{3 6 1}}=19$

So, it is rational.

(iii) $\sqrt{21}$

$\sqrt{21}=\sqrt{3} \times \sqrt{7}=4.58257 \ldots$

It is an irrational number.

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

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

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

(ix) $3.040040004 \ldots$ is an irrational number because it is a non-terminating, non-repeating decimal.

(x) $2.356565656 \ldots$ is a rational number because it is repeating.

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