Write the lowest rationalizing factor of :

(i) $5\sqrt{2}$

(ii) $\sqrt{24}$

(iii) $\sqrt{5} - 3$

(iv) $7 - \sqrt{7}$

(v) $\sqrt{18} - \sqrt{50}$

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

(vii) $\sqrt{13} + 3$

(i) Given,

$\Rightarrow 5\sqrt{2}$

Rationalizing,

$\Rightarrow 5\sqrt{2} \times \sqrt{2} \\[1em] \Rightarrow 10.$

Hence, lowest rationalizing factor of $5\sqrt{2} = \sqrt{2}$.

(ii) Given,

$\Rightarrow \sqrt{24} \\[1em] \Rightarrow 2\sqrt{6}.$

Rationalizing,

$\Rightarrow 2\sqrt{6} \times \sqrt{6} \\[1em] \Rightarrow 12.$

Hence, lowest rationalizing factor of $\sqrt{24} = \sqrt{6}$.

(iii) Given,

$\Rightarrow \sqrt{5} - 3$

Rationalizing,

$\Rightarrow (\sqrt{5} - 3) \times (\sqrt{5} + 3) \\[1em] \Rightarrow (\sqrt{5})^2 - 3^2 \\[1em] \Rightarrow 5 - 9 \\[1em] \Rightarrow -4.$

Hence, lowest rationalizing factor of $\sqrt{5} - 3 = \sqrt{5} + 3$.

(iv) Given,

$\Rightarrow 7 - \sqrt{7}$

Rationalizing,

$\Rightarrow (7 - \sqrt{7}) \times (7 + \sqrt{7}) \\[1em] \Rightarrow (7)^2 - (\sqrt{7})^2 \\[1em] \Rightarrow 49 - 7 \\[1em] \Rightarrow 42.$

Hence, lowest rationalizing factor of $7 - \sqrt{7} = 7 + \sqrt{7}$.

(v) Given,

$\Rightarrow \sqrt{18} - \sqrt{50}\\[1em] \Rightarrow 3\sqrt{2} - 5\sqrt{2}$

Rationalizing,

$\Rightarrow (3\sqrt{2} - 5\sqrt{2}) \times (\sqrt{2}) \\[1em] \Rightarrow (6 - 10) \\[1em] \Rightarrow -4$

Hence, lowest rationalizing factor of $\sqrt{18} - \sqrt{50} = \sqrt{2}$.

(vi) Given,

$\Rightarrow \sqrt{5} - \sqrt{2}$

Rationalizing,

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

Hence, lowest rationalizing factor of $\sqrt{5} - \sqrt{2} = \sqrt{5} + \sqrt{2}$.

(vii) Given,

$\Rightarrow \sqrt{13} + 3$

Rationalizing,

$\Rightarrow (\sqrt{13} + 3)(\sqrt{13} - 3) \\[1em] \Rightarrow (\sqrt{13})^2 - 3^2 \\[1em] \Rightarrow 13 - 9 \\[1em] \Rightarrow 4.$

Hence, lowest rationalizing factor of $\sqrt{13} + 3 = \sqrt{13} - 3$.