基本介紹
![負指數冪](/img/3/a96/wZwpmL3MzM1IDO3EjMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL2AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/9/31e/wZwpmLxITM5IDO2IjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLwgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
在指數法則 中,如果 ,則就產生了 負指數冪。
定義負指數冪等於把冪指數變號後所得的冪的倒數。也就是
![負指數冪](/img/e/026/wZwpmLwIDM2YTO5UjM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL2EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![負指數冪](/img/4/ad2/wZwpmL0AjM3kzMwIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/0/f1f/wZwpmLygzMwcDMzgTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4EzL3MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
應該知道,負指數冪也是不能用正整指數冪的意義來解釋的。也就是說“ ”不能認為是“ 個 相乘”的意思。另外在定義中規定底數不得為零,其原因是和零指數冪的定義是一樣的。
![負指數冪](/img/4/2b3/wZwpmLwIjN1MzM1cTN1ATN0UTMyITNykTO0EDMwAjMwUzL3UzLxEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/4/ad1/wZwpmL0cTN4cjNxQTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![負指數冪](/img/6/8b3/wZwpmL4MDN4kTMyUDM0kTO0UTMyITNykTO0EDMwAjMwUzL1AzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/1/633/wZwpmL4UzMzcDNyIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLyQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/1/93f/wZwpmL1EzM0gDMxMDO4EDN0UTMyITNykTO0EDMwAjMwUzLzgzL2UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/2/b06/wZwpmL3UDO0UDOxEDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLxQzLzczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![負指數冪](/img/4/2b3/wZwpmLwIjN1MzM1cTN1ATN0UTMyITNykTO0EDMwAjMwUzL3UzLxEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/1/93f/wZwpmL1EzM0gDMxMDO4EDN0UTMyITNykTO0EDMwAjMwUzLzgzL2UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/1/f00/wZwpmLyYjMyADO1czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
在 中規定, ,這是因為 產生於 , 當 時, ,我們知道0是不能作除數的, 所以 中,當 時, 這是沒有意義的 。
相關概念
冪
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/a/250/wZwpmLzUDMwUjN3QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
n個 相乘的積稱為“ 的n次冪”或“ 的n次乘方”記作 , 是底數,n是指數。這裡n可以是分數、負數,分別稱為“分指數冪”、“負指數冪”,也可以是任意實數或複數。
分指數冪
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/9/b12/wZwpmLwYDN2YDM2MTO0AzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLxYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/9/b12/wZwpmLwYDN2YDM2MTO0AzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLxYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
當冪的指數為分數時,稱為“分指數冪”。正數 的 次冪( 是既約正分數)定義為 的m次冪的n次算術根,就是:
![負指數冪](/img/c/ccf/wZwpmL0EzMxIDN1czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL1QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
乘方
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/a/250/wZwpmLzUDMwUjN3QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1)n個 相乘的積 稱為 的n次“乘方”,參見“冪”。
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/a/250/wZwpmLzUDMwUjN3QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2)從 求 的運算,稱為“乘方”。
正整數指數冪
![負指數冪](/img/5/0d7/wZwpmL2YTM5UTNyEDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/a/e3f/wZwpmL1AzN1kDO3ATMwEDN0UTMyITNykTO0EDMwAjMwUzLwEzL4UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/a/e3f/wZwpmL1AzN1kDO3ATMwEDN0UTMyITNykTO0EDMwAjMwUzLwEzL4UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![負指數冪](/img/9/e87/wZwpmL4gjN2kTOxIjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLwMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
一般地, 叫作 的 次冪, 叫作冪的 底數, 叫作冪的 指數,並且規定 。我們注意到在 的n次冪定義中,n是正整數,因此通常又把它稱為 正整數指數冪 。
容易驗證,正整數指數冪的運算滿足如下法則:
![負指數冪](/img/b/148/wZwpmLwYzN4cDM3QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLyUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(1) ;
![負指數冪](/img/6/2e9/wZwpmLzAjM5YDM0EzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxMzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2) ;
![負指數冪](/img/8/70d/wZwpmL3cTNxIDO3QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLwUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(3) ;
![負指數冪](/img/0/7ce/wZwpmLyYTO5ADN5EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL4UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(4) 。
負整數指數冪
![負指數冪](/img/b/e07/wZwpmL4MjMwkzN4MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL4IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
在法則(3)中規定了 ,如果取消這個限制,就需要討論下面兩種情形:
![負指數冪](/img/3/c0a/wZwpmL1UzM5QDN5AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL1AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
❶當 時,冪的商有如下運算:
![負指數冪](/img/3/5ef/wZwpmLyIDNyYDN4czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL0AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
依照法則(3)則有
![負指數冪](/img/2/f76/wZwpmLxUjN2IzN5EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/5/7ae/wZwpmL4MjMwgTM5UjM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL2AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
即
這就說明當指數為負整數時,冪的值是有意義的。此時規定
![負指數冪](/img/4/b40/wZwpmLxMjMycTN0YTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2EzL1QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![負指數冪](/img/1/fc5/wZwpmL0ATMxgDM0QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL1IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
叫作 負整數指數冪。
零指數冪
![負指數冪](/img/e/d35/wZwpmL0gzMwIjMxYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL3gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
❷當 時,冪的商有如下運算:
![負指數冪](/img/1/de8/wZwpmLzAjN3UzN1czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL4EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![負指數冪](/img/b/605/wZwpmL3QDM4gTN0gTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4EzLzgzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![負指數冪](/img/0/6e2/wZwpmL2IzMykDN3EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLzUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
且 故
這說明當指數為零時,冪的值是有意義的。此時規定
![負指數冪](/img/2/0c3/wZwpmL1YTMxgzN1QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/4/dce/wZwpmLxYzN4cjM0EjMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLyQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![負指數冪](/img/5/2f4/wZwpmLxgzM0kjM5cTO4kzM0UTMyITNykTO0EDMwAjMwUzL3kzLwgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
叫作 零指數冪,又叫零次冪。但是 是無意義的。
正整數指數冪、負整數指數冪、零指數冪統稱為整數指數冪。正整數指數冪的運算法則對整數指數冪仍然是成立的。特別地,有
![負指數冪](/img/2/0c3/wZwpmL1YTMxgzN1QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![負指數冪](/img/4/2e4/wZwpmL2YTN1ATM2IjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
整數指數冪的運算法則
同上所述,容易驗證,正整數指數冪的運算滿足如下法則:
![負指數冪](/img/b/148/wZwpmLwYzN4cDM3QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLyUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(1) ;
![負指數冪](/img/6/2e9/wZwpmLzAjM5YDM0EzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLxMzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2) ;
![負指數冪](/img/8/70d/wZwpmL3cTNxIDO3QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLwUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(3) ;
![負指數冪](/img/0/7ce/wZwpmLyYTO5ADN5EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL4UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(4) 。
![負指數冪](/img/1/511/wZwpmL2ITN3ETN1AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(5) ;
註:①這些運算性質在整數指數範圍內仍然適用。
![負指數冪](/img/b/617/wZwpmLyAjNwkTO2QzMwgDM1UTM1QDN5MjM5ADMwAjMwUzL0MzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![負指數冪](/img/0/7eb/wZwpmL1UzM3EjNzIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLzYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![負指數冪](/img/4/ad1/wZwpmL0cTN4cjNxQTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![負指數冪](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
②任何不等於零的數的 (n為正整數)次冪,等於這個數的n次冪的倒數,即 ( ,n為正整數)。在這兩個冪的意義中,強調底數 都不等於零,否則無意義。
③學習了零指數冪和負整數指數冪後,正整數指數冪的運算性質可以推廣到整數指數幕的範圍 。
冪的運算法則
當指數概念擴充到任意實數之後,冪的運算法則可合併為 :
![負指數冪](/img/a/64b/wZwpmLzYDN2YTN1YzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLyQzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1) ;
![負指數冪](/img/9/fa3/wZwpmL2gDO4ITOxYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL2gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(2) ;
![負指數冪](/img/6/8ef/wZwpmL4YDM5MTN1IDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLyEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(3) 。