Question

√((√12-√8) (√3+√2) /(5+√24))

Answer 1

Please see the attachment

Answer 2

$\sqrt{\dfrac{(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})}{(5+\sqrt{24})} }=\sqrt{10-4\sqrt{6}}$

Step-by-step explanation:

We have,

$\sqrt{\dfrac{(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})}{(5+\sqrt{24})} }$

To find, $\sqrt{\dfrac{(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})}{(5+\sqrt{24})} }=?$

=$\sqrt{\dfrac{(2\sqrt{3}-2\sqrt{2})(\sqrt{3}+\sqrt{2})}{(5+\sqrt{24})} }$

$=\sqrt{\dfrac{2(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}{(5+\sqrt{24})} }$

$=\sqrt{\dfrac{2(\sqrt{3}^2-\sqrt{2}^2)}{(5+\sqrt{24})}}$

[ ∵ $a^{2} -b^{2}=(a+b)(a-b)$]

$=\sqrt{\dfrac{2(3-2)}{(5+\sqrt{24})}}$

$=\sqrt{\dfrac{2}{(5+\sqrt{24})}}$

Rationalising numerator and denominator, we get

$=\sqrt{\dfrac{2}{(5+\sqrt{24})}\times \dfrac{5-\sqrt{24}}{5-\sqrt{24}} }$

$=\sqrt{\dfrac{2(5-\sqrt{24})}{(5^2-\sqrt{24}^2)}$

$=\sqrt{\dfrac{10-2\sqrt{24}}{(25-24)}$

$=\sqrt{\dfrac{10-2\sqrt{24}}{1}$

=$\sqrt{10-4\sqrt{6}}$

Hence, $\sqrt{\dfrac{(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})}{(5+\sqrt{24})} }=\sqrt{10-4\sqrt{6}}$