1 . 复数的乘方:实数集
中正整数指数的运算律,在复数集
中仍然成立,只不过是要把运算的结果写成复数的代数形式罢了.即若
,m,n是正整数,则
①
; ②
;③
;④
.
复数的除法运算法则:复数的除法,实质上就是分母“实数化”——将分母化为实数,即分子、分母同乘以分母的共轭复数.类似于以前所学的分母“有理化”.于是,我们得到,当
,且
时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ce4f6edb26ddfc86b73b0c9443c37e.png)
______________ .
的乘方的性质及其应用:在计算
的高次幂的值时,常常利用
,
简化运算.如计算
时,先将其表示成
与
的积,再将
看成是
,于是得到![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60f086009af2e86309dd8688f1f97c27.png)
___________ .
设
,利用复数的四则运算法则,可以得到
具有下面的性质:
①![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63fdfb84247dffc8fcc6678d34e00dd2.png)
_________ ; ②
; ③
; ④![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12c3c68fb14a6c63bf4d0869c71e6513.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/933093b52cca887f597cbe22a5467b11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f015ed8e497b4394053ddd19683a98f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe639d6800b34618d379dd1fc07d74af.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/336ad2461f5bc92f6f4e6f7d981983c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb3d8634bc85df0a4a04550cca29231c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f193a2071132d8466a161cefcc50c4a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a129424a8b6a52dfd0c6568a266669f.png)
复数的除法运算法则:复数的除法,实质上就是分母“实数化”——将分母化为实数,即分子、分母同乘以分母的共轭复数.类似于以前所学的分母“有理化”.于是,我们得到,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3b40d453ac56a449af2c33e31ff49c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bff71e0a1fb212d900f59491248bf3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ce4f6edb26ddfc86b73b0c9443c37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf922fbceea012b857449a80079adc8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0abea11dc55beb34183e5542443c98c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33c7d11e443ee9aeca7e45e6d31afe1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/064b0e7b2494d7341339be6a80961158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f7ad7ce320f21c4b1f60bc8e381239b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5faa2af08cf50fa3efada313a694d7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c167774cf4c5c67c0af0a5b1cb5a6528.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5faa2af08cf50fa3efada313a694d7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/048c2a7371541a685e4d66b38b5c282c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60f086009af2e86309dd8688f1f97c27.png)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afc2753abf67f40c8070fda7538cda6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63fdfb84247dffc8fcc6678d34e00dd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0377e5c91846b3a0e71f4cc03ca1c9c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080397a23c2bc5b9d87261e3795d5d68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12c3c68fb14a6c63bf4d0869c71e6513.png)
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