电子课本网 第120页

第120页

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$ \begin{aligned}解:原式&= \sqrt{x²+y²}-x \sqrt{x²+y²}+\frac{1}{y}·y\sqrt {x²+y² } \\ &=(2-x) \sqrt{x²+y²} \\ \end{aligned}$
$解:∵a^{2} -3a+1=0,$
$∴a-3+\frac {1}{a}=0,$
$∴a+\frac {1}{a}=3,$
$∴(a+\frac {1}{a})^{2} =9,$
$∴a^{2} +\frac {1}{a^{2} }+2=9,$
$∴a^{2} +\frac {1}{a^{2} }=7,$
$∴\sqrt {a^{2} +\frac {1}{a^{2} }+5} =\sqrt {7+5}=2\sqrt {3} .\ $
$解:错在对 \sqrt{(x-2)²}的化简,正确计算:$
$∵x=2-\sqrt{3}, ∴x-2<0.\ $
$ \begin{aligned} ∴原式&=\frac{(x-2)^{2} }{x-2} +\frac{\sqrt{(x-2)²}}{x(x-2)}\ \\ &=x-2-\frac{x-2}{x(x-2)}=x-2-\frac{1}{x}\ \\ &=2-\sqrt{3}-2-\frac{1}{2-\sqrt{3}}=-\sqrt {3} -(2+\sqrt {3} )\ \\ &= -\sqrt{3}-2-\sqrt{3}=-2-2\sqrt{3}. \\ \end{aligned}$
$解:∵m=\sqrt{2}+2\sqrt{3},n=2\sqrt{2}-\sqrt{3},$
$∴m+n=3\sqrt{2}+\sqrt{3},m-n=3\sqrt{3}-\sqrt{2},$
$ \begin{aligned}∴原式&=\frac{1}{3\sqrt{2}+\sqrt{3}}-\frac{1}{3\sqrt{3}-\sqrt{2}} \\ &=\frac{3\sqrt{2}-\sqrt{3}}{15}-\frac{3\sqrt{3}+\sqrt{2}}{25} \\ &=\frac{\sqrt{2}}{5}-\frac{\sqrt{3}}{15}-\frac{3\sqrt{3}}{25}-\frac{\sqrt{2}}{25} \\ &=\frac {4}{25}\sqrt 2-\frac {14}{75}\sqrt 3 \\ \end{aligned}$
$ \begin{aligned}解:原式&=( \frac{2a}{a-1}-\frac {a}{a-1})÷a \\ &=\frac{a}{a-1}· \frac{1}{a} \\ &=\frac{1}{a-1}.\ \\ \end{aligned}$
$当a=\sqrt {2} +1时,$
$原式=\frac{1}{a-1}=\frac{1}{\sqrt {2} +1-1}=\frac{1}{\sqrt {2} }=\frac{\sqrt {2} }{2}.$
$ \begin{aligned} 解:原式&=( \frac{3x+y}{x²-y²}-\frac{2x}{x²-y²})÷\frac{2}{x²y-xy²} \\ &=\frac{3x+y-2x}{(x-y)(x+y)} ·\frac{xy(x-y)}{2} \\ &=\frac{x+y}{(x-y)(x+y)} ·\frac {xy(x- y)}{2} \\ &=\frac{xy}{2}, \\ \end{aligned}$
$当x= \sqrt{3}+1,y=\sqrt{3}时,$
$原式=\frac{\sqrt{3}(\sqrt{3}+1)}{2}=\frac{3+\sqrt {3} }{2}.$