`ĐKXĐ:x\ge0,x\ne1`

`D=({3\sqrt{x}}/{\sqrt{x}-1}-{1}/{\sqrt{x}+1}-3).{\sqrt{x}+1}/{\sqrt{x}+2}`

`=[{3\sqrt{x}(\sqrt{x}+1)}/{(\sqrt{x}+1)(\sqrt{x}-1)}-{\sqrt{x}-1}/{(\sqrt{x}+1)(\sqrt{x}-1)}-{3(\sqrt{x}+1)(\sqrt{x}-1)}/{(\sqrt{x}+1)(\sqrt{x}-1)}].{\sqrt{x}+1}/{\sqrt{x}+2}` 

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

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

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

`={2\sqrt{x}+4}/{(\sqrt{x}+1)(\sqrt{x}-1)}.{\sqrt{x}+1}/{\sqrt{x}+2}` 

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

`={2}/{\sqrt{x}-1}` 

Vậy với `x\ge0,x\ne1` thì `D={2}/{\sqrt{x}-1}` 

`ĐKXĐ:a\ge0,b\ge0,a\neb`

`E=({\sqrt{a}+\sqrt{b}}/{\sqrt{a}-\sqrt{b}}-{4\sqrt{ab}}/{a-b}).[{a\sqrt{a}+b\sqrt{b}}/{\sqrt{ab}-(a+b)}]`

`=[{\sqrt{a}+\sqrt{b}}/{\sqrt{a}-\sqrt{b}}-{4\sqrt{ab}}/{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}].{(\sqrt{a}+\sqrt{b})(a-\sqrt{ab}+b)}/{\sqrt{ab}-a-b}`

`=[{(\sqrt{a}+\sqrt{b})(\sqrt{a}+\sqrt{b})}/{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}-{4\sqrt{ab}}/{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}].{(\sqrt{a}+\sqrt{b})(a-\sqrt{ab}+b)}/{-a+\sqrt{ab}-b}`

`={(\sqrt{a}+\sqrt{b})(\sqrt{a}+\sqrt{b})-4\sqrt{ab}}/{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}.{(\sqrt{a}+\sqrt{b})(a-\sqrt{ab}+b)}/{-(a-\sqrt{ab}+b)}`

`={(\sqrt{a}+\sqrt{b})^2-4\sqrt{ab}}/{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}.{(\sqrt{a}+\sqrt{b})(a-\sqrt{ab}+b)}/{-(a-\sqrt{ab}+b)}`

`={a+2\sqrt{ab}+b-4\sqrt{ab}}/{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}.{(\sqrt{a}+\sqrt{b})(a-\sqrt{ab}+b)}/{-(a-\sqrt{ab}+b)}`

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

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

`=-(\sqrt{a}-\sqrt{b})`

`=\sqrt{b}-\sqrt{a}`

Vậy với `a\ge0,b\ge0,a\neb` thì `E=\sqrt{b}-\sqrt{a}`

Hướng dẫn trả lời:

Bài 4:

a) `D = ({3sqrt{x}}/{sqrt{x} - 1} - {1}/{sqrt{x} + 1} - 3)cdot{sqrt{x} + 1}/{sqrt{x} + 2}` với `x ≥ 0; x ne 1`

`= ({3sqrt{x}cdot(sqrt{x} + 1)}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)} - {sqrt{x} - 1}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)} - {3cdot(sqrt{x} + 1)cdot(sqrt{x} - 1)}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)})cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {3sqrt{x}cdot(sqrt{x} + 1) - (sqrt{x} - 1) - 3cdot(sqrt{x} + 1)cdot(sqrt{x} - 1)}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {3sqrt{x}cdot\sqrt{x} + 3sqrt{x}cdot1 - sqrt{x} + 1 - 3cdot[(sqrt{x})^2 - 1^2]}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {3x + 3sqrt{x} - sqrt{x} + 1 - 3cdot(x - 1)}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {3x + 3sqrt{x} - sqrt{x} + 1 - 3x + 3}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {(3x - 3x) + (3sqrt{x} - sqrt{x}) + (1 + 3)}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {2sqrt{x} + 4}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {2cdot(sqrt{x} + 2)}/{(sqrt{x} + 1)cdot(sqrt{x} - 1)}cdot{sqrt{x} + 1}/{sqrt{x} + 2}`

`= {2}/{sqrt{x} - 1}`

b) `E = ({sqrt{a} + sqrt{b}}/{sqrt{a} - sqrt{b}} - {4sqrt{ab}}/{a - b})cdot({asqrt{a} + bsqrt{b}}/{sqrt{ab} - (a + b)})`

với `a ≥0; b ≥ 0; a ne b`

`= ({sqrt{a} + sqrt{b}}/{sqrt{a} - sqrt{b}} - {4sqrt{ab}}/{(sqrt{a})^2 - (sqrt{b})^2})cdot({(sqrt{a})^3 + (sqrt{b})^3}/{sqrt{ab} - a - b})`

`= ({sqrt{a} + sqrt{b}}/{sqrt{a} - sqrt{b}} - {4sqrt{ab}}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})})cdot{(sqrt{a} + sqrt{b})cdot[(sqrt{a})^2 - sqrt{a}cdot\sqrt{b} + (sqrt{b})^2]}/{- (a - sqrt{ab} + b)}`

`= ({(sqrt{a} + sqrt{b})^2}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})} - {4sqrt{ab}}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})})cdot-{(sqrt{a} + sqrt{b})cdot(a - sqrt{ab} + b)}/{a - sqrt{ab} + b}`

`= {(sqrt{a} + sqrt{b})^2 - 4sqrt{ab}}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})}cdot-(sqrt{a} + sqrt{b})`

`= {(sqrt{a})^2 + 2cdot\sqrt{a}cdot\sqrt{b} + (sqrt{b})^2 - 4sqrt{ab}}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})}cdot-(sqrt{a} + sqrt{b})`

`= {a + 2\sqrt{ab} + b - 4sqrt{ab}}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})}cdot-(sqrt{a} + sqrt{b})`

`= {a - 2\sqrt{ab} + b}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})}cdot-(sqrt{a} + sqrt{b})`

`= {(sqrt{a})^2 - 2cdot\sqrt{a}cdot\sqrt{b} + (sqrt{b})^2}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})}cdot-(sqrt{a} + sqrt{b})`

`= {(sqrt{a} - sqrt{b})^2}/{(sqrt{a} + sqrt{b})cdot(sqrt{a} - sqrt{b})}cdot-(sqrt{a} + sqrt{b})`

`= - (sqrt{a} - sqrt{b})`

`= - sqrt{a} + sqrt{b}`