Find the square of :

(i) $\dfrac{3\sqrt{5}}{5}$

(ii) $\sqrt{3} + \sqrt{2}$

(iii) $\sqrt{5} - 2$

(iv) $3 + 2\sqrt{5}$

(i) Squaring,

$\Rightarrow \Big(\dfrac{3\sqrt{5}}{5}\Big)^2 \\[1em] \Rightarrow \dfrac{3\sqrt{5} \times 3\sqrt{5}}{5 \times 5} \\[1em] \Rightarrow \dfrac{45}{25} \\[1em] \Rightarrow \dfrac{9}{5} \\[1em] \Rightarrow 1\dfrac{4}{5}.$

Hence, square of $\dfrac{3\sqrt{5}}{5} = 1\dfrac{4}{5}$.

(ii) Squaring,

$\Rightarrow (\sqrt{3} + \sqrt{2})^2 \\[1em] \Rightarrow (\sqrt{3})^2 + (\sqrt{2})^2 + 2 \times \sqrt{3} \times \sqrt{2} \\[1em] \Rightarrow 3 + 2 + 2\sqrt{6} \\[1em] \Rightarrow 5 + 2\sqrt{6}.$

Hence, square of $\sqrt{3} + \sqrt{2} = 5 + 2\sqrt{6}$.

(iii) Squaring,

$\Rightarrow (\sqrt{5} - 2)^2 \\[1em] \Rightarrow (\sqrt{5})^2 + (2)^2 - 2 \times \sqrt{5} \times 2 \\[1em] \Rightarrow 5 + 4 - 4\sqrt{5} \\[1em] \Rightarrow 9 - 4\sqrt{5}.$

Hence, square of $\sqrt{5} - 2 = 9 - 4\sqrt{5}$.

(iv) Squaring,

$\Rightarrow (3 + 2\sqrt{5})^2 \\[1em] \Rightarrow (3)^2 + (2\sqrt{5})^2 + 2 \times 3 \times 2\sqrt{5} \\[1em] \Rightarrow 9 + 20 + 12\sqrt{5} \\[1em] \Rightarrow 29 + 12\sqrt{5}.$

Hence, square of $3 + 2\sqrt{5} = 29 + 12\sqrt{5}$.