Question

Identify whether the following numbers are rational or irrational.
(i) $3+\sqrt{3}$
(ii) $(4+\sqrt{2})-(4-\sqrt{3})$
(iii) $\frac{\sqrt{18}}{2\sqrt{2}}$
(iv) $\sqrt{19}-(2+\sqrt{19})$
(v) $\frac{2}{\sqrt{3}}$
(vi) $\sqrt{12}\times \sqrt{3}$.

Answer 1

(i) $3+\sqrt{3}$
$3$ is a rational number and $\sqrt{3}$ is irrational. Hence, $3+\sqrt{3}$ is irrational.
(ii) $(4+\sqrt{2})-(4-\sqrt{3})$
$=4+\sqrt{2}-4+\sqrt{3}=\sqrt{2}+\sqrt{3}$, is irrational.
(iii) $\frac{\sqrt{18}}{2\sqrt{2}}=\frac{\sqrt{9\times 2}}{2\sqrt{2}}=\frac{\sqrt{9}\times \sqrt{2}}{2\sqrt{2}}=\frac{3}{2}$, is rational.
(iv) $\sqrt{19}-(2+\sqrt{19})=\sqrt{19}-2-\sqrt{19}=-2,$ is rational.
(v) $\frac{2}{\sqrt{3}}$ here $2$ is rational and $\sqrt{3}$ is irrational. Hence, $\frac{2}{\sqrt{3}}$ is irrational.
(vi) $\sqrt{12}\times \sqrt{3}=\sqrt{12\times 3}=\sqrt{36}=6$, is rational.