Answer:
√6
Step-by-step explanation:
We have to find the value of the expression [tex]\frac{\sqrt{7+\sqrt{24} } }{2} +\frac{\sqrt{7-\sqrt{24} } }{2}[/tex].
Let, [tex]A = \frac{\sqrt{7+\sqrt{24} } }{2} +\frac{\sqrt{7-\sqrt{24} } }{2}[/tex]
⇒ [tex]A^{2} = \frac{7+\sqrt{24} }{4} +\frac{7-\sqrt{24} }{4}+ 2\times \frac{\sqrt{7+\sqrt{24} } }{2} \times \frac{\sqrt{7-\sqrt{24} } }{2}[/tex] {Squaring both sides. Since (a + b)² = a² + b² + 2ab}
⇒ [tex]A^{2} = \frac{7+\sqrt{24}+7-\sqrt{24} }{4} + 2 \times \frac{\sqrt{49-24} }{4}[/tex] {Since (a + b)(a - b) = a² - b²}
⇒ [tex]A^{2}= \frac{7}{2} +2 \times \frac{5}{4}[/tex]
⇒ [tex]A^{2} = \frac{7}{2} +\frac{5}{2} = 6[/tex]
⇒ A = √6 (Answer) {Neglecting the negative root as the original expression is positive}
