State which of the following numbers are irrational :

$\begin{matrix} \text{(i)} & \sqrt{\dfrac{4}{9}}, -{\dfrac{3}{70}},\sqrt{\dfrac{7}{25}},\sqrt{\dfrac{16}{5}} \\[1.5em] \text{(ii)} & -{\sqrt{\dfrac{2}{49}}}, {\dfrac{3}{200}},\sqrt{\dfrac{25}{3}},-{\sqrt{\dfrac{49}{16}}} \\[1.5em] \end{matrix}$

$\text{(i) } \sqrt{\dfrac{4}{9}} = \dfrac{2}{3} \\[1.5em] -\dfrac{3}{70} = -\dfrac{3}{70} \\[1.5em] \sqrt{\dfrac{7}{25}} = \dfrac{\sqrt{7}}{5} \\[1.5em] \sqrt{\dfrac{16}{5}} = \dfrac{4}{\sqrt{5}} \\[1.5em]$

∴ $\bold{\sqrt{\dfrac{7}{25}}}$ and $\bold{\sqrt{\dfrac{16}{5}}}$ are irrational numbers as they cannot be written in the form $\dfrac{{p}}{q}$ where p and q are integers.

$\bold{\sqrt{\dfrac{4}{9}}}$ and $\bold{- \dfrac{3}{70}}$ are rational numbers and they can be written in the form $\dfrac{{p}}{q}$ where p and q are integers .

$\text{(ii) } - \sqrt{\dfrac{2}{49}} = - \dfrac{\sqrt{2}}{7} \\[1.5em] \dfrac{3}{200} = \dfrac{3}{200} \\[1.5em] \sqrt{\dfrac{25}{3}} = \dfrac{5}{\sqrt{3}} \\[1.5em] -\sqrt{\dfrac{49}{16}} = -\dfrac{7}{{4}} \\[1.5em]$

∴ $\bold{- \sqrt{\dfrac{2}{49}}}$ and $\bold{\sqrt{\dfrac{25}{3}}}$ are irrational numbers as they cannot be written in the form $\dfrac{{p}}{q}$ where p and q are integers.

$-\bold{\sqrt{\dfrac{49}{16}}}$ and $\bold{\dfrac{3}{200}}$ are rational numbers and they can be written in the form $\dfrac{{p}}{q}$ where p and q are integers.