**Question:**

Classify the following numbers as rational or irrational with justification

(i) $\sqrt{196}$

(ii) $3 \sqrt{18}$

(iii) $\sqrt{\frac{9}{27}}$

(iv) $\frac{\sqrt{28}}{\sqrt{343}}$

(v) $-\sqrt{0.4}$

(vi) $\frac{\sqrt{12}}{\sqrt{75}}$

(vii) $0.5918$

(viii) $(1+\sqrt{5})-(4+\sqrt{5})$

(ix) $10.124124 \ldots$

(x) $1.010010001 \ldots$

Thinking Process

To classify, use the definition a rational number is in the form of $\mathrm{p} / \mathrm{q}$, where $p$ and $q$ are integers and $q \neq 0$ and otherwise it is an irrational number.

**Solution:**

(i) $\sqrt{196}=\sqrt{(14)^{2}}=14$

Hence, it is a rational number.

(ii) $3 \sqrt{18}=3 \sqrt{(3)^{2} \times 2}=3 \times 3 \sqrt{2}=9 \sqrt{2}$

Hence, it is an irrational number.

(iii) $\sqrt{\frac{9}{27}}=\sqrt{\frac{9}{9 \times 3}}=\frac{1}{\sqrt{3}}$

Hence, it is an irrational number, because $\sqrt{3}$ is an irrational number.

(iv) $\frac{\sqrt{28}}{\sqrt{343}}=\frac{\sqrt{2 \times 2 \times 7}}{\sqrt{7 \times 7 \times 7}}=\frac{2 \sqrt{7}}{7 \sqrt{7}}=\frac{2}{7}$

Hence, it is a rational number.

(v) $-\sqrt{0.4}=-\sqrt{\frac{4}{10}}=-\frac{2}{\sqrt{10}}$

Hence, it is a quotient of rational and irrational numbers, so it is an irrational number.

(vi) $\frac{\sqrt{12}}{\sqrt{75}}=\frac{\sqrt{4 \times 3}}{\sqrt{25 \times 3}}=\frac{\sqrt{4} \sqrt{3}}{\sqrt{25} \sqrt{3}}=\frac{2}{5}$

Hence, it is a rational number.

(vii) $0.5918$, it is a number with terminating decimal, so it can be written in the form of $\frac{p}{q}$,

where $q \neq 0, p$ and $q$ are integers.

Hence, it is a rational number.

(viii) $(1+\sqrt{5})-(4+\sqrt{5})=1-4+\sqrt{5}-\sqrt{5}=-3$

Hence, it is a rational number.

(ix) $10.124124$ is a number with non-terminating recurring decimal expansion.

Hence, it is a rational number.

(x) $1.010010001$ is a number with non-terminating non-recurring decimal expansion.

Hence, it is a irrational number.