Question

Question 4
Classify the following numbers as rational or irrational with justification.
$\left(i\right)\sqrt{196}\left(ii\right)3\sqrt{18}\left(iii\right)\sqrt{\frac{9}{27}}\left(iv\right)\frac{\sqrt{28}}{343}\left(v\right)-\sqrt{0.4}\left(vi\right)\frac{\sqrt{12}}{\sqrt{75}}$
(vii) 0.5918
(viii) $(1+\sqrt{5})-(4+\sqrt{5})$
(ix) 10.124124 ....
(x) 1.010010001....
To classify, use the definition a rational number is in the form of $\frac{p}{q}$, where p and q are integers and $q\ne 0$ and otherwise it is an irrational number.

Answer 1

(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.

$\left(iii\right)\sqrt{\frac{9}{27}}=\sqrt{\frac{9}{9\times 3}}=\frac{1}{\sqrt{3}}$
Hence, it is an irrational number, because $\sqrt{3}$ is an irrational number.

(iv) $\sqrt{\frac{28}{343}}=\frac{\sqrt{2\times 2\times 7}}{\sqrt{7\times 7\times 7}}=\frac{2\sqrt{7}}{7\sqrt{7}}=\frac{2}{7}$
Hence, it is a rational number.

$\left(v\right)-\sqrt{0.4}=-\sqrt{\frac{4}{10}}=-\frac{2}{\sqrt{10}}$
Hence, it is an irrational number, because $\sqrt{10}$ is an irrational number.

$\left(vi\right)\frac{\sqrt{12}}{\sqrt{75}}=\frac{\sqrt{4\times 3}}{\sqrt{25\times 3}}=\frac{\sqrt{4}\sqrt{3}}{\sqrt{25}\sqrt{3}}=\frac{2}{5}$
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 $q\ne 0$, p and q are integers. Hence, it is a rational number.

$\left(viii\right)(1+\sqrt{5})-(4+\sqrt{5})=1-4+\sqrt{5}-\sqrt{5}=-3$
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 an irrational number.