24.

a. $\sqrt{\dfrac{2}{3}}$

$=\dfrac{\sqrt 2}{\sqrt 3}$

$=\dfrac{\sqrt 2 . \sqrt 3}{3}$

$=\dfrac{\sqrt 6}{3}$

b. $\sqrt{\dfrac{x^2}{5}} (x \geq 0)$

$=\dfrac{\sqrt{x^2}}{\sqrt 5}$

$=\dfrac{\sqrt 5 .|x|}{5}$

$=\dfrac{x\sqrt 5}{5} (x \geq 0)$

c. $\sqrt{\dfrac{3}{x}} (x> 0)$
$=\dfrac{\sqrt 3}{\sqrt x}$

$=\dfrac{\sqrt 3 . \sqrt x}{x}$

$=\dfrac{\sqrt{3x}}{x}$

d. $\sqrt{x^2 - \dfrac{x^2}{7}} (x<0)$
$= \sqrt{\dfrac{7x^2 - x^2}{7}}$

$= \dfrac{\sqrt{6x^2}}{\sqrt 7}$

$=\dfrac{\sqrt{6}.|x|.\sqrt 7}{7}$

$=\dfrac{-\sqrt{42}x}{7} (x< 0)$

25.

+)

$\dfrac{2}{\sqrt 6 - \sqrt 5}$

$=\dfrac{2(\sqrt 6 + \sqrt 5)}{(\sqrt 6 - \sqrt 5)(\sqrt 6 + \sqrt 5)}$

$=\dfrac{2\sqrt 6 + 2\sqrt 5}{1}$

$=2\sqrt 6 + 2\sqrt 5$

+)

$\dfrac{3}{\sqrt{10} + \sqrt 7}$

$=\dfrac{3(\sqrt{10} - \sqrt 7)}{(\sqrt{10} - \sqrt 7)(\sqrt{10} + \sqrt 7)}$

$=\dfrac{3(\sqrt{10} - \sqrt 7)}{3}$

$=\sqrt{10} - \sqrt{7}$

+)

$\dfrac{1}{\sqrt x - \sqrt y} (x \geq 0 ; y \geq 0; x \neq y)$

$=\dfrac{\sqrt x + \sqrt y}{(\sqrt x - \sqrt y)(\sqrt x + \sqrt y)}$

$=\dfrac{\sqrt x + \sqrt y}{x - y}$

+)

$\dfrac{2ab}{\sqrt a - \sqrt b} (a \geq 0 ; b \geq 0; a \neq b)$

$=\dfrac{2ab(\sqrt a + \sqrt b}{(\sqrt a - \sqrt b)(\sqrt a + \sqrt b)}$

$=\dfrac{2ab(\sqrt a + \sqrt b)}{a - b}$

26.

a. $\dfrac{2}{\sqrt 3 - 1} - \dfrac{2}{\sqrt 3 +1}$

$=\dfrac{2(\sqrt 3 + 1) - 2(\sqrt 3 - 1)}{(\sqrt 3- \sqrt 1)(\sqrt 3 + \sqrt 1)}$

$=\dfrac{2\sqrt 3 + 2 - 2\sqrt 3 + 2}{3 - 1}$

$=\dfrac{4}{2}$

$=2$

b. $\dfrac{5}{12(2\sqrt 5 + 3\sqrt 2)} - \dfrac{5}{12(2\sqrt 5 - 3\sqrt 2)}$

$=\dfrac{5(2\sqrt 5 - 3\sqrt 2) - 5(2\sqrt 5 + 3\sqrt 2)}{12(2\sqrt 5 - 3\sqrt 2)(2\sqrt 5 + 3\sqrt 2)}$

$=\dfrac{10\sqrt 5 - 15\sqrt 2 - 10\sqrt 5 - 15\sqrt 2 }{12(20 - 18)}$

$=\dfrac{-30\sqrt 2}{42}$

$=\dfrac{-5\sqrt 2}{7}$

c. $\dfrac{5 + \sqrt 5}{5 - \sqrt 5} + \dfrac{5 -\sqrt 5}{5 + \sqrt 5}$ 

$=\dfrac{\sqrt 5(\sqrt 5 + 1)}{\sqrt  5(\sqrt 5 - 1)}  + \dfrac{\sqrt 5(\sqrt 5 - 1)}{\sqrt  5(\sqrt 5 +1)}$

$=\dfrac{\sqrt 5 + 1}{\sqrt 5 - 1} + \dfrac{\sqrt 5 - 1}{\sqrt 5 + 1}$

$=\dfrac{(\sqrt 5 + 1)^2 + (\sqrt 5 - 1)^2}{(\sqrt 5 + 1)(\sqrt 5 -1)}$

$=\dfrac{6 + 2\sqrt 5 + 6 - 2\sqrt 5}{5 - 1}$

$=\dfrac{12}{4}$

$=3$

d. $\dfrac{\sqrt 3}{\sqrt{3 + 1} - 1} - \dfrac{\sqrt 3}{\sqrt{3+1} +1}$

$= \dfrac{\sqrt 3}{2 - 1} - \dfrac{\sqrt 3}{2 +1}$

$=\dfrac{\sqrt 3}{1} - \dfrac{\sqrt 3}{3}$

$=\sqrt 3 - \dfrac{1}{\sqrt 3}$

$=\dfrac{3 - 1}{\sqrt 3}$

$=\dfrac{2}{\sqrt 3}$

$=\dfrac{2\sqrt 3}{3}$