Question

`\frac{ \sqrt{7 + √(24) } } 2{ + } \frac{ \sqrt{7 - √(24) } }{2}`
​

Answer 1

√6

Explanation:

We have to find the value of the expression
`\frac{\sqrt{7+√(24) } }{2} +\frac{\sqrt{7-√(24) } }{2}`.

Let,
`A = \frac{\sqrt{7+√(24) } }{2} +\frac{\sqrt{7-√(24) } }{2}`

⇒
`A^(2) = (7+√(24) )/(4) +(7-√(24) )/(4)+ 2* \frac{\sqrt{7+√(24) } }{2} * \frac{\sqrt{7-√(24) } }{2}` {Squaring both sides. Since (a + b)² = a² + b² + 2ab}

⇒
`A^(2) = (7+√(24)+7-√(24)  )/(4) + 2 * (√(49-24) )/(4)` {Since (a + b)(a - b) = a² - b²}

⇒
`A^(2)= (7)/(2)  +2 * (5)/(4)`

⇒
`A^(2) = (7)/(2) +(5)/(2) = 6`

⇒ A = √6 (Answer) {Neglecting the negative root as the original expression is positive}