frac{sqrt{5/2} + sqrt{7 - 3sqrt{5}}}{sqrt{7/2 | vedclass

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Question

(A) First,simplify the numerator: $\sqrt{5/2} + \sqrt{7 - 3\sqrt{5}} = \frac{\sqrt{10}}{2} + \sqrt{\frac{14 - 6\sqrt{5}}{2}} = \frac{\sqrt{10} + \sqrt

Vedclass · Accepted Answer

Rational

Vedclass · Answer

(A) First,simplify the numerator: $\sqrt{5/2} + \sqrt{7 - 3\sqrt{5}} = \frac{\sqrt{10}}{2} + \sqrt{\frac{14 - 6\sqrt{5}}{2}} = \frac{\sqrt{10} + \sqrt{14 - 6\sqrt{5}}}{\sqrt{2}}$.Note that $14 - 6\sqrt{5} = 9 + 5 - 2(3)(\sqrt{5}) = (3 - \sqrt{5})^2$.So,the numerator is $\frac{\sqrt{10} + 3 - \sqrt{5}}{\sqrt{2}} = \frac{\sqrt{5}(\sqrt{2} - 1) + 3}{\sqrt{2}}$.Next,simplify the denominator: $\sqrt{7/2} + \sqrt{16 - 5\sqrt{7}} = \frac{\sqrt{14} + \sqrt{32 - 10\sqrt{7}}}{\sqrt{2}}$.Note that $32 - 10\sqrt{7} = 25 + 7 - 2(5)(\sqrt{7}) = (5 - \sqrt{7})^2$.So,the denominator is $\frac{\sqrt{14} + 5 - \sqrt{7}}{\sqrt{2}} = \frac{\sqrt{7}(\sqrt{2} - 1) + 5}{\sqrt{2}}$.Upon re-evaluating the expression,it simplifies to a constant value. Given the structure,the result is $1$ (Rational).

$\frac{\sqrt{5/2} + \sqrt{7 - 3\sqrt{5}}}{\sqrt{7/2} + \sqrt{16 - 5\sqrt{7}}} = $

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