Question

Help me in this question quickly pleaseeeeee

Answer 1

$let \: \: x = \sqrt{ \frac{5}{4} + \sqrt{ \frac{3}{2} } } \: \: and \\ y = \sqrt{ \frac{5}{4} - \sqrt{ \frac{3}{2} } } \\ \\ then \: \: {x}^{2} + {y}^{2} \\ = (\frac{5}{4} + \sqrt{ \frac{3}{2} } )+ (\frac{5}{4} - \sqrt{ \frac{3}{2} }) = \frac{5}{2} \\ \\ xy = (\sqrt{ \frac{5}{4} + \sqrt{ \frac{3}{2} } } )(\sqrt{ \frac{5}{4} - \sqrt{ \frac{3}{2} } } ) \\ = \sqrt{(\frac{5}{4} + \sqrt{ \frac{3}{2} })(\frac{5}{4} - \sqrt{ \frac{3}{2} })} \\ = \sqrt{( \frac{25}{16} - \frac{3}{2} )} = \frac{1}{4} \\ \\ {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \\ = \frac{5}{2} + 2 \times \frac{1}{4} = 3 \\ so \: \: x + y = \sqrt{3} \\ \\ therefore \\ \sqrt{ \frac{5}{4} + \sqrt{ \frac{3}{2} } } + \sqrt{ \frac{5}{4} - \sqrt{ \frac{3}{2} } } = \sqrt{3}$