Toán Các phép biến đổi về căn thức-10

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a. [tex]\frac{(\sqrt{x^{2}+4}-2)(\sqrt{x^{2}+4}+2)(x+\sqrt{x}+1)\sqrt{x-2\sqrt{x}+1}}{x(x\sqrt{x}-1)}[/tex] với x>0 và x#1

b.[tex]\frac{(\sqrt{x^{2}+1}-1)(\sqrt{x^{2}+1}+1)(x-\sqrt{x}+1)\sqrt{x+2\sqrt{x}+1}}{x(x\sqrt{x}+1)}[/tex] với x>0

[tex]d. (\frac{\sqrt{a}-2}{\sqrt{a}+2}-\frac{\sqrt{a}+2}{\sqrt{a}-2})(\sqrt{a}-\frac{4}{\sqrt{a}})[/tex] với a>0, a#4

[tex]f. \frac{x^{2}-\sqrt{x}}{x+\sqrt{x}+1}-\frac{x^{2}+\sqrt{x}}{x-\sqrt{x}+1}[/tex]+x+1 với x>0

[tex]g. (\frac{2+\sqrt{x}}{x+2\sqrt{x}+1}-\frac{\sqrt{x}-2}{x-1})(\frac{x\sqrt{x}+x-\sqrt{x}-1}{\sqrt{x}})[/tex] với x>0

[tex]h. \left [ 1:(1-\frac{\sqrt{a}}{1+\sqrt{a}}) \right ](\frac{1}{\sqrt{a}-1}-\frac{2\sqrt{a}}{(a+1)(\sqrt{a}-1)})[/tex]

[tex]i. (\frac{\sqrt{a^{3}}-\sqrt{b^{3}}}{\sqrt{a}-\sqrt{b}}-\sqrt{ab}):\frac{\frac{1}{a^{2}}-\frac{1}{b^{2}}}{\frac{1}{a}-\frac{1}{b}}[/tex]
[tex]j. \left [ (\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}).\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{1}{x}+\frac{1}{y} \right ]:\frac{\sqrt{x^{3}}+y\sqrt{x}+x\sqrt{y}+\sqrt{y^{3}}}{\sqrt{x^{3}y}+\sqrt{xy^{3}}}[/tex]
[tex]k. (\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1):(1-\frac{\sqrt{} xy+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1})[/tex]

 

[Nữ Thần Mặt Trăng]

a. [tex]\frac{(\sqrt{x^{2}+4}-2)(\sqrt{x^{2}+4}+2)(x+\sqrt{x}+1)\sqrt{x-2\sqrt{x}+1}}{x(x\sqrt{x}-1)}[/tex] với x>0 và x#1

b.[tex]\frac{(\sqrt{x^{2}+1}-1)(\sqrt{x^{2}+1}+1)(x-\sqrt{x}+1)\sqrt{x+2\sqrt{x}+1}}{x(x\sqrt{x}+1)}[/tex] với x>0

[tex]d. (\frac{\sqrt{a}-2}{\sqrt{a}+2}-\frac{\sqrt{a}+2}{\sqrt{a}-2})(\sqrt{a}-\frac{4}{\sqrt{a}})[/tex] với a>0, a#4

[tex]f. \frac{x^{2}-\sqrt{x}}{x+\sqrt{x}+1}-\frac{x^{2}+\sqrt{x}}{x-\sqrt{x}+1}[/tex]+x+1 với x>0

[tex]g. (\frac{2+\sqrt{x}}{x+2\sqrt{x}+1}-\frac{\sqrt{x}-2}{x-1})(\frac{x\sqrt{x}+x-\sqrt{x}-1}{\sqrt{x}})[/tex] với x>0

[tex]h. \left [ 1:(1-\frac{\sqrt{a}}{1+\sqrt{a}}) \right ](\frac{1}{\sqrt{a}-1}-\frac{2\sqrt{a}}{(a+1)(\sqrt{a}-1)})[/tex]

[tex]i. (\frac{\sqrt{a^{3}}-\sqrt{b^{3}}}{\sqrt{a}-\sqrt{b}}-\sqrt{ab}):\frac{\frac{1}{a^{2}}-\frac{1}{b^{2}}}{\frac{1}{a}-\frac{1}{b}}[/tex]
[tex]j. \left [ (\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}).\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{1}{x}+\frac{1}{y} \right ]:\frac{\sqrt{x^{3}}+y\sqrt{x}+x\sqrt{y}+\sqrt{y^{3}}}{\sqrt{x^{3}y}+\sqrt{xy^{3}}}[/tex]

a) $\dfrac{(\sqrt{x^{2}+4}-2)(\sqrt{x^{2}+4}+2)(x+\sqrt{x}+1)\sqrt{x-2\sqrt{x}+1}}{x(x\sqrt{x}-1)}$
$=\dfrac{x^2(x+\sqrt x+1)|\sqrt{x}-1|}{x(\sqrt{x}-1)(x+\sqrt{x}+1)}=\dfrac{x|\sqrt x-1|}{\sqrt x-1}=\left\{\begin{matrix}x \ \text{nếu} \ x>1\\ -x \ \text{nếu} \ 0<x<1\end{matrix}\right.$
b) $\dfrac{(\sqrt{x^{2}+1}-1)(\sqrt{x^{2}+1}+1)(x-\sqrt{x}+1)\sqrt{x+2\sqrt{x}+1}}{x(x\sqrt{x}+1)}$
$=\dfrac{x^2(x-\sqrt x+1)|\sqrt x+1|}{x(\sqrt x+1)(x-\sqrt x+1)}=\dfrac{x(\sqrt x+1)}{\sqrt x+1}=x$
d) $(\dfrac{\sqrt{a}-2}{\sqrt{a}+2}-\dfrac{\sqrt{a}+2}{\sqrt{a}-2})(\sqrt{a}-\dfrac{4}{\sqrt{a}})=\dfrac{(\sqrt a-2)^2-(\sqrt a+2)^2}{x-4}.\dfrac{a-4}{\sqrt a}=\dfrac{-8\sqrt a}{\sqrt a}=-8$
f) $\dfrac{x^{2}-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{x^{2}+\sqrt{x}}{x-\sqrt{x}+1}+x+1$
$=\dfrac{\sqrt{x}(\sqrt{x}-1)(x+\sqrt{x}+1)}{x+\sqrt{x}+1}-\dfrac{\sqrt{x}(\sqrt{x}+1)(x-\sqrt{x}+1)}{x-\sqrt{x}+1}+x+1$
$=x-\sqrt{x}-x-\sqrt{x}+x+1=x-2\sqrt{x}+1$
g) $(\dfrac{2+\sqrt{x}}{x+2\sqrt{x}+1}-\dfrac{\sqrt{x}-2}{x-1})(\dfrac{x\sqrt{x}+x-\sqrt{x}-1}{\sqrt{x}})$
$=\dfrac{(\sqrt{x}+2)(\sqrt{x}-1)-(\sqrt{x}-2)(\sqrt x+1)}{(\sqrt{x}-1)(\sqrt{x}+1)^2}.\dfrac{(\sqrt{x}-1)(\sqrt{x}+1)^2}{\sqrt{x}}$
$=\dfrac{x+\sqrt x-2-x+\sqrt{x}+2}{\sqrt{x}}=\dfrac{2\sqrt{x}}{\sqrt x}=2$
h) $\left [ 1: (1-\dfrac{\sqrt{a}}{1+\sqrt{a}}) \right ](\dfrac{1}{\sqrt{a}-1}-\dfrac{2\sqrt{a}}{(a+1)(\sqrt{a}-1)})$
$=(\sqrt a+1).\dfrac{a+1-2\sqrt a}{(a+1)(\sqrt a-1)}=\dfrac{(\sqrt a+1)(\sqrt a-1)^2}{(a+1)(\sqrt a-1)}=\dfrac{a-1}{a+1}$
i) $(\dfrac{\sqrt{a^{3}}-\sqrt{b^{3}}}{\sqrt{a}-\sqrt{b}}-\sqrt{ab}):\dfrac{\dfrac{1}{a^{2}}-\dfrac{1}{b^{2}}}{\dfrac{1}{a}-\dfrac{1}{b}}=(a+\sqrt{ab}+b-\sqrt{ab})(\dfrac1a+\dfrac1b)=\dfrac{(a+b)^2}{ab}$
j) $\left [ (\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}).\dfrac{2}{\sqrt{x}+\sqrt{y}}+\dfrac{1}{x}+\dfrac{1}{y} \right ]:\dfrac{\sqrt{x^{3}}+y\sqrt{x}+x\sqrt{y}+\sqrt{y^{3}}}{\sqrt{x^{3}y}+\sqrt{xy^{3}}}$
$=\dfrac{2\sqrt{xy}+x+y}{xy}:\dfrac{(\sqrt{x}+\sqrt{y})(x+y)}{\sqrt{xy}(x+y)}=\dfrac{(\sqrt x+\sqrt y)^2}{xy}.\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}=\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}}$