State whether the following numbers are rational or not :

(i) $(2 + \sqrt{2})^2$

(ii) $(3 - \sqrt{3})^2$

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

(iv) $(\sqrt{3} - \sqrt{2})^2$

(i) Given,

$\Rightarrow (2 + \sqrt{2})^2 \\[1em] \Rightarrow 4 + 2 + 4\sqrt{2} \\[1em] \Rightarrow 6 + 4\sqrt{2} \\[1em] \Rightarrow 6 + 5.658…. \\[1em] \Rightarrow 11.658….$

which is irrational.

Hence, $(2 + \sqrt{2})^2$ is not a rational number.

(ii) Given,

$\Rightarrow (3 - \sqrt{3})^2 \\[1em] \Rightarrow 9 + 3 - 6\sqrt{3} \\[1em] \Rightarrow 12 - 6\sqrt{3} \\[1em] \Rightarrow 12 - 10.392….. \\[1em] \Rightarrow 1.607……$

which is irrational.

Hence, $(3 - \sqrt{3})^2$ is not a rational number.

(iii) Given,

$\Rightarrow (5 + \sqrt{5})(5 -\sqrt{5}) \\[1em] \Rightarrow 25 - 5\sqrt{5} + 5\sqrt{5} - 5 \\[1em] \Rightarrow 20$

which is rational.

Hence, $(5 + \sqrt{5})(5 -\sqrt{5})$ is a rational number.

(iv) Given,

$\Rightarrow (\sqrt{3} - \sqrt{2})^2 \\[1em] \Rightarrow 3 + 2 - 2 \times \sqrt{3} \times \sqrt{2} \\[1em] \Rightarrow 5 - 2\sqrt{6} \\[1em] \Rightarrow 5 - 2 \times 2.449… \\[1em] \Rightarrow 5 - 4.898….. \\[1em] \Rightarrow 0.101…..$

which is irrational.

Hence, $(\sqrt{3} - \sqrt{2})^2$ is not a rational number.