1 . 我们知道形如
,
的数可以化简,其化简的目的主要是把原数分母中的无理数化为有理数.如:
,
.这样的化简过程叫做分母有理化.我们把
叫做
的有理化因式,
叫做
的有理化因式.利用有理化因式,可以得到如下结论:①
;②设有理数a,b满足
,则
;③
;④已知
,则
;以上结论正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf30225033ed65ba166c6f25ffa2ff52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46d903100e4258ef742d56efdeeb156.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e13086a7dd306655c58926773a80a60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6631a036244034b582312289cd0ce6c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab46ea0cba2d06283fae3d864a2329e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47cfe4e08c06bde245e58aa22485044c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51a4f6d6a05b6fb6e3160a665d13d806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f538b8da9794174c7b0bf67c54ac8ea0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289295fad0d1c11fc74470e8bb0c6be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c3cc9f76c7b756e3a77811e3b475bad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cc1f48f40d6aa2219122ded96bf8ae3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b9d406a76e15e4216175219db0c9b34.png)
A.①③ | B.①② | C.①④ | D.③④ |
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2 . 计算
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ef3bfe73e5b623736c50171e42a224.png)
(2)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ef3bfe73e5b623736c50171e42a224.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f49900d0abd350b3ae192d11a61dcc9b.png)
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3 . 计算:
(1)
;
(2)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701611ab767255ff49f6cacf348770cc.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a96855c56f4af27b8224a9127229ed.png)
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4 . 【阅读】我们将
与
称为一对“对偶式”,
因为
,所以构造“对偶式”,再将其相乘可以有效地将
和
中的“
”去掉,于是二次根式的除法可以这样计算:如![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c78d9b11fb443069fb7e09ea86822ce.png)
.像这样,通过分子、分母同乘一个式子把分母中的根号化去,叫做分母有理化.
根据以上材料,理解并运用材料提供的方法,解答下列问题:
(1)对偶式
与
之间的关系是____________;
A.互为相反数 B.绝对值相等 C.互为倒数
(2)已知
,
,化简
,
;
(3)解方程:
.
[提示:令
,
].
(4)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a989c4fbf0ff34cedb365d2dda47f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/175d0c99e866f1db915462a91b356ef3.png)
因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50036f42d68bcb48ee54bf6a8e76856f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9928ba39cdd5878aced506d3dbb9fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b87033f1efe85e559a9a3efc985be9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aa0dba76fbef65983ebf043218ea5e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c78d9b11fb443069fb7e09ea86822ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3755d7e08b98c26303f2f61de7e7ddd2.png)
根据以上材料,理解并运用材料提供的方法,解答下列问题:
(1)对偶式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab46ea0cba2d06283fae3d864a2329e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47cfe4e08c06bde245e58aa22485044c.png)
A.互为相反数 B.绝对值相等 C.互为倒数
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69e4636f3b6eb54711c4410fa831f623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/add3e1ba560da4c0904e016eb85940b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(3)解方程:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e95857b9fcde7b45e59ef3d498e456.png)
[提示:令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab49c882cfe6fc298a74b8dbfa5cd815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50036f42d68bcb48ee54bf6a8e76856f.png)
(4)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75fd7e405a9928ae2642a15050b1446.png)
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5 . 对于任意不相等的两个数
,定义一种运算“*”如下
,如
,计算:
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62ff2912fd8d93b6e692936d95b727c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4763aa41ec20b7a7de7ebdae8f74d1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7475616eba61e570c1f71e4fd1f1c68e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b60f888ff701252b6f38961167156263.png)
A.2 | B.3 | C.4 | D.6 |
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6 . 已知
,
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc12fed7a6f4777c0fccc896e624281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c4bb3296249d8b1ca8416d3c4a0bf75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf9bb992b634d2ed82c1e65c090bfc27.png)
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7 . 材料阅读:二次根式的运算中,经常会出现诸如
的计算,将分母转化为有理数,这就是“分母有理化”
;
.
类似地,将分子转化为有理数,就称为“分子有理化”
;
.
根据上述知识,请你解答下列问题:
(1)化简
;
(2)比较
与
的大小,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bacf860bd03668eea97fc63ca7a9504.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74d42aa863fea1d89edc6b59106ee43e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64bba8670c57e5ef5fe2c3da39494fbf.png)
类似地,将分子转化为有理数,就称为“分子有理化”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2b4b3477db96c93095aa745cf757cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fbecda6d216d5f73ccab8913bca3288.png)
根据上述知识,请你解答下列问题:
(1)化简
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f3d3ca1d50810cffde511af72ec9972.png)
(2)比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36351e0b64226f965f092fc9f4e788b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a672590b05847a76b3eb3ddc5282468.png)
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浙江省金华市义乌市群星外国语学校2023-2024学年八年级下学期3月份数学试题山东省济宁市任城区济宁学院附属中学2023-2024学年八年级下学期4月期中考试数学试题(已下线)清单05 二次根式 全章复习(3个考点梳理+11种题型+10类型)-2023-2024学年八年级数学下学期期末考点大串讲(苏科版)(已下线)暑假作业09 二次根式及其运算(知识梳理+11大题型+拓展突破)-【暑假分层作业】2024年八年级数学暑假培优练(苏科版)
8 . 下列运算正确的是( )
A.![]() | B.![]() |
C.![]() | D.![]() |
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9 . 计算:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96bcde2f5ad899f66f65dfb24f739936.png)
__ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96bcde2f5ad899f66f65dfb24f739936.png)
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10 . 计算:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bd0fec6b4210be3770933323a23f0f4.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/398ca52855447b0384591ecbb79b2017.png)
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