Rút gọn:
a) $\frac{1}{3+2\sqrt{2}}$ + $\frac{4\sqrt{2}-4}{2-\sqrt{2}}$
b) `\sqrt{x}`. ($\frac{1}{\sqrt{x}+3}$ - $\frac{1}{3-\sqrt{x}}$ )
c) $\frac{x-2\sqrt{xy} +y}{x-y}$ + $\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}$
d) $\frac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}$ +$\frac{4}{\sqrt{3}+1}$ - `\sqrt{(3\sqrt{3}-5)^2}`
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Đáp án+Giải thích các bước giải:
`a)`
`\frac{1}{3+2\sqrt{2}}+\frac{4\sqrt{2}-4}{2-\sqrt{2}}`
`=\frac{3-2\sqrt{2}}{(3+2\sqrt{2}).(3-2\sqrt{2})}+\frac{4.(\sqrt{2}-1)}{\sqrt{2}.(\sqrt{2}-1)}`
`=\frac{3-2\sqrt{2}}{3^{2}-(2\sqrt{2})^{2}}+\frac{4}{\sqrt{2}}`
`=\frac{3-2\sqrt{2}}{9-8}+\frac{4\sqrt{2}}{(\sqrt{2})^{2}}`
`=3-2\sqrt{2}+\frac{4\sqrt{2}}{2}`
`=3-2\sqrt{2}+2\sqrt{2}`
`=3+(2\sqrt{2}-2\sqrt{2})`
`=3+0`
`=3`
`b)`
`\sqrt{x}.(\frac{1}{\sqrt{x}+3}-\frac{1}{3-\sqrt{x}})` `(đk:x\ge0;x\ne9)`
`=\sqrt{x}.[\frac{\sqrt{x}-3}{(\sqrt{x}-3).(\sqrt{x}+3)}+\frac{\sqrt{x}+3}{(\sqrt{x}-3).(\sqrt{x}+3)}]`
`=\sqrt{x}.[\frac{\sqrt{x}-3+\sqrt{x}+3}{(\sqrt{x}-3).(\sqrt{x}+3)}]`
`=\frac{\sqrt{x}.2\sqrt{x}}{(\sqrt{x}-3).(\sqrt{x}+3)}`
`=\frac{2x}{(\sqrt{x})^{2}-3^{2}}`
`=\frac{2x}{x-9}`
`c)`
`\frac{x-2\sqrt{xy}+y}{x-y}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}` `(đk:x>0;y>0)`
`=\frac{(\sqrt{x})^{2}-2.\sqrt{x}.\sqrt{y}+(\sqrt{y})^{2}}{(\sqrt{x})^{2}-(\sqrt{y})^{2}}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}`
`=\frac{(\sqrt{x}-\sqrt{y})^{2}}{(\sqrt{x}-\sqrt{y}).(\sqrt{x}+\sqrt{y})}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}`
`=\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}`
`=\frac{\sqrt{x}-\sqrt{y}+2\sqrt{y}}{\sqrt{x}+\sqrt{y}}`
`=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}}`
`=1`
`d)`
`\frac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}+\frac{4}{\sqrt{3}+1}-\sqrt{(3\sqrt{3}-5)^{2}}`
`=\frac{\sqrt{3}.(\sqrt{5}-\sqrt{4})}{\sqrt{5}-2}+\frac{4.(\sqrt{3}-1)}{(\sqrt{3}+1).(\sqrt{3}-1)}-|3\sqrt{3}-5|`
`=\frac{\sqrt{3}.(\sqrt{5}-2)}{\sqrt{5}-2}+\frac{4.(\sqrt{3}-1)}{(\sqrt{3})^{2}-1^{2}}-(3\sqrt{3}-5)`
Vì `3\sqrt{3}-5>0=>|3\sqrt{3}-5|=3\sqrt{3}-5`
`=\sqrt{3}+\frac{4.(\sqrt{3}-1)}{2}-3\sqrt{3}+5`
`=-2\sqrt{3}+5+2.(\sqrt{3}-1)`
`=-2\sqrt{3}+5+2\sqrt{3}-2`
`=3`
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Đáp án:
Giải thích các bước giải:
`a) 1/(3+2sqrt(2))+(4sqrt(2)-4)/(2-sqrt(2))`
`= (3-2sqrt(2))/(3^2 -(2sqrt(2))^2)+(4(sqrt(2)-1))/(sqrt(2)(sqrt(2)-1))`
`=3-2sqrt(2) + 4/(sqrt(2))`
`= 3-2sqrt(2)+ (4sqrt(2))/2`
`= 3-2sqrt(2)+2sqrt(2)`
`= 3`
`b) sqrt(x).(1/(sqrt(x)+3)- 1/(3-sqrt(x)))`
`(đk: x \ne 9; x >= 0)`
`= sqrt(x). ((sqrt(x)-3)/(x-9)+ (sqrt(x)+3)/(x-9))`
`= sqrt(x). (2sqrt(x))/(x-9)`
`= (2x)/(x-9)`
`c) (x-2sqrt(xy)+y)/(x-y)+(2sqrt(y))/(sqrt(x)+sqrt(y))`
`(đk: x >= 0; y >= 0; x \ne y )`
`= ((sqrt(x)-sqrt(y))^2)/((sqrt(x)-sqrt(y)(sqrt(x)+sqrt(y)))+(2sqrt(y))/(sqrt(x)+sqrt(y))`
`= (sqrt(x)-sqrt(y))/(sqrt(x)+sqrt(y)+(2sqrt(y))/(sqrt(x)+sqrt(y))`
`= (sqrt(x)+sqrt(y))/(sqrt(x)+sqrt(y))`
`=1`
`d) (sqrt(15)-sqrt(12))/(sqrt(5)-2)+ 4/(sqrt(3)+1)- sqrt((3sqrt(3)-5)^2)`
`= (sqrt(3)(sqrt(5)-2))/(sqrt(5)-sqrt(2))+ (4(sqrt(3)-1))/(3-1) - |3sqrt(3)-5|`
`= sqrt(3) + (4(sqrt(3)-1))/2 - 3sqrt(3) +5`
`= -2sqrt(3) + 2sqrt(3) -2 +5`
`= 3`
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