`24.`

`-` ĐKXĐ : `{ (a > 0) , (b > 0) , (a != b) :}`

`a)P = [ 1 / (\sqrt{a} + \sqrt{b}) + (3\sqrt{ab}) / (a\sqrt{a} + b\sqrt{b}) ] . [ 1 / (\sqrt{a} - \sqrt{b}) - (3\sqrt{ab}) / (a\sqrt{a} - b\sqrt{b}) ] : (a - b) / (a + \sqrt{ab} + b)`

`= [ (a - \sqrt{ab} + b + 3\sqrt{ab}) / ((\sqrt{a} + \sqrt{b})(a - \sqrt{ab} + b)) ] . [ (a + \sqrt{ab} + b - 3\sqrt{ab}) / ((\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)) ] . (a + \sqrt{ab} + b) / (a - b) `

`= [ (a + 2\sqrt{ab} + b) / (a\sqrt{a} + b\sqrt{b}) ] . [ (a - 2\sqrt{ab} + b) / (a\sqrt{a} - b\sqrt{b}) ] . (a + \sqrt{ab} + b) / (a - b) `

`= [ (\sqrt{a} + \sqrt{b})^2 / ((\sqrt{a} + \sqrt{b})(a - \sqrt{ab} + b)) ] . [ (\sqrt{a} - \sqrt{b})^2 / ((\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)) ] . (a + \sqrt{ab} + b) / (a - b) `

`= [ (\sqrt{a} + \sqrt{b}) / (a - \sqrt{ab} + b) ] . [ (\sqrt{a} - \sqrt{b}) / (a + \sqrt{ab} + b) ] . (a + \sqrt{ab} + b) / (a - b) `

`= ((\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})) / ((a - \sqrt{ab} + b)(a - b)) `

`= (a - b) / ((a - \sqrt{ab} + b)(a - b)) `

`= 1 / (a - \sqrt{ab}+b) `

`b)` Thay `a = 16` và `b = 4` có :

`P = 1 / (16 - \sqrt{16 . 4} + 4) = 1 / (16 - 8 + 4) = 1 / 12`

`25.`

`-` ĐKXĐ : `{ (a >= 0) , (a != 1) , (a != 1/4) :} `

`a)P = 1 + [ ((2\sqrt{a} - 1)(\sqrt{a} + 1)) / ((1 - \sqrt{a})(1 + \sqrt{a})) - (\sqrt{a}(2\sqrt{a} - 1)(\sqrt{a} + 1)) / ((1 - \sqrt{a})(1 + \sqrt{a} + a)) ] . (\sqrt{a}(\sqrt{a} - 1)) / (2\sqrt{a} - 1)`

`= 1 + [ (2\sqrt{a} - 1) / (1 - \sqrt{a}) - (\sqrt{a}(2\sqrt{a} - 1)(\sqrt{a} + 1)) / ((1 - \sqrt{a})(1 + \sqrt{a} + a)) ] . [ -(\sqrt{a}(1 - \sqrt{a})) / (2\sqrt{a} - 1) ] `

`= 1 - (2\sqrt{a} - 1) / (1 - \sqrt{a}) . (\sqrt{a}(1 - \sqrt{a})) / (2\sqrt{a} - 1) + (\sqrt{a}(2\sqrt{a} - 1)(\sqrt{a} + 1)) / ((1 - \sqrt{a})(1 + \sqrt{a} + a)) . (\sqrt{a}(1 - \sqrt{a})) / (2\sqrt{a} - 1) `

`= 1 - \sqrt{a} + (a(\sqrt{a} + 1)) / (1 + \sqrt{a} + a) `

`= ((1 - \sqrt{a})(1 + \sqrt{a} + a) + a\sqrt{a} + a) / (1 + \sqrt{a} + a) `

`= (1 - a\sqrt{a} + a\sqrt{a} + a) / (1 + \sqrt{a} + a)`

`= (a + 1) / (1 + \sqrt{a} + a)`

`b)P = \sqrt{6} / (1 + \sqrt{6}) `

`-> (a + 1) / (1 + \sqrt{a} + a) = \sqrt{6} / (1 + \sqrt{6}) `

`-> (a + 1)(1 + \sqrt{6}) = \sqrt{6}(1 + \sqrt{a} + a) `

`-> a + 1 + a\sqrt{6} + \sqrt{6} = \sqrt{6} + \sqrt{6a} + a\sqrt{6} `

`-> a + 1 = \sqrt{6a} `

`-> (a + 1)^2 = 6a `

`->a^2 + 2a + 1 = 6a`

`-> a^2 - 4a + 1 = 0`

`-> (a^2 - 4a + 4) - 3 = 0`

`-> (a - 2)^2 = 3`

`-> a - 2 = \sqrt{3}` hoặc `a - 2 = -\sqrt{3}`

`-> a = 2 + \sqrt{3}` hoặc `a = 2 - \sqrt{3}` (thỏa mãn)

`c)` Xét `P - 2/3= (a + 1) / (1 + \sqrt{a} + a) - 2/3`

`= (3a + 3 - 2 - 2\sqrt{a} - 2a) / (3(1 + \sqrt{a} + a)) `

`= (a - 2\sqrt{a} + 1) / (3(1 + \sqrt{a} + a)) `

`= (\sqrt{a} - 1)^2 / (3(1 + \sqrt{a} + a))`

Có `{((\sqrt{a} - 1)^2 >= 0AAa),(3(1 + \sqrt{a} + a) > 0AAa >= 0):}`

`-> P - 2/3 >= 0 `

`-> P >= 2/3` (đpcm)