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    精英家教网 > 初中数学 > 题目详情
    阅读并按要求完成各题．
    （1）在化简
    		7
    -2
    		10
    的过程中，小王和小李的化简结果不一样：
    小王的化简过程如下：
    原式=
    		2-2
    		2×5
    +5
    =
    		(
    		2
    )    2    -2
    		2
    •
    		5
    +(
    		5
    )    2
    =
    		(
    		2
    -
    		5
    )    2
    =
    		2
    -
    		5
    ．
    小李的化简过程如下：
    原式=
    		(
    		2
    )    2    -2
    		2
    •
    		5
    +(
    		5
    )    2
    =
    		(
    		2
    -
    		5
    )    2
    =
    		5
    -
    		2
    ．
    请判断谁的化简结果正确，并说明理由．
    （2）已知x=
    		6-2
    		5
    ，求(
    1
    x-2
    +
    1
    x+2
    )    •
    x2-4
    2(x-1)
    的值．（结果保留根号）
    分析：（1）根据二次根式的性质得到
    		(
    		2
    -
    		5)2
    =|
    		2
    -
    		5
    |=
    		5
    -
    		2
    ，由此可判断小李的化简结果正确．
    （2）理由（1）的方法可化简x得到x=
    		5
    -1，再对分式进行通分、约分得到原式=
    x
    x-1
    ，然后把x的值代入后解分母有理化即可．
    解答：解：（1）小李的化简结果正确．理由如下：
    		(
    		2
    -
    		5)2
    =|
    		2
    -
    		5
    |=
    		5
    -
    		2
    ；
    （2）x=
    		5-2
    		5
    +1
    =
    		(
    		5)2-2
    		5
    +1
    =
    		(
    		5
    -1)2
    =
    		5
    -1，
    原式=
    x+2+x-2
    (x-2)(x+2)
    •
    (x+2)(x-2)
    2(x-1)

    =
    x
    x-1
    ，
    当x=
    		5
    -1时，原式=
    		5
    -1
    		5
    -1-1
    =3+
    		5
    ．
    点评：本题考查了二次根式的化简求值：先化简再代入求值；二次根式运算的最后，注意结果要化到最简二次根式，二次根式的乘除运算要与加减运算区分，避免互相干扰．也考查了分式的化简求值．
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