# 微積分學/代數

 ← 基礎知識 微積分學 函數 → 代數

## 四則運算

### 加法

• ${\displaystyle a+0=a}$
• ${\displaystyle a+(-a)=0}$
• 交換律：${\displaystyle a+b=b+a}$
• 結合律：${\displaystyle (a+b)+c=a+(b+c)}$

### 減法

• 定義：${\displaystyle a-b=a+(-b)}$

### 乘法

• ${\displaystyle a\times 1=a}$
• ${\displaystyle a\neq 0}$ 時，${\displaystyle a\times {\frac {1}{a}}=1}$
• 交換律：${\displaystyle a\times b=b\times a}$
• 結合律：${\displaystyle (a\times b)\times c=a\times (b\times c)}$
• 分配律：${\displaystyle a\times (b+c)=(a\times b)+(a\times c)}$

### 除法

• 定義：當${\displaystyle b\neq 0}$ 時，${\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}$

 ${\displaystyle {\frac {(x+2)(x+3)}{x+3}}}$ ${\displaystyle =\left[(x+2)\times (x+3)\right]\times {\frac {1}{x+3}}}$ （除法定義） ${\displaystyle =(x+2)\times \left[(x+3)\times {\frac {1}{x+3}}\right]}$ （乘法結合律） ${\displaystyle =(x+2)\times 1,\qquad x\neq -3}$ ${\displaystyle =x+2,\qquad x\neq -3}$

## 指數運算

${\displaystyle n}$ 為正整數，則記${\displaystyle a^{n}}$ ${\displaystyle a}$ 底數）的${\displaystyle n}$ 指數）次方，即

${\displaystyle a^{n}=a\cdot a\cdot a\cdots a}$ ${\displaystyle n}$ 次）

${\displaystyle a\neq 0}$ ，則${\displaystyle a^{0}=1}$

${\displaystyle -n}$ 為負整數，則${\displaystyle a^{-n}={\frac {1}{a^{n}}}}$

${\displaystyle a^{n}a^{m}=a^{n+m}}$  ${\displaystyle 3^{6}\times 3^{9}=3^{15}}$
${\displaystyle {\frac {a^{n}}{a^{m}}}=a^{n-m}}$  ${\displaystyle {\frac {x^{3}}{x^{2}}}=x^{1}=x}$
${\displaystyle (a^{n})^{m}=a^{n\cdot m}}$  ${\displaystyle (x^{4})^{5}=x^{20}}$
${\displaystyle (ab)^{n}=a^{n}b^{n}}$  ${\displaystyle (3x)^{5}=3^{5}x^{5}}$
${\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}}$  ${\displaystyle \left({\frac {7}{3}}\right)^{3}={\frac {7^{3}}{3^{3}}}}$

${\displaystyle 144^{\frac {5}{3}}=(2^{4}\cdot 3^{2})^{\frac {5}{3}}=2^{\frac {20}{3}}\cdot 3^{\frac {10}{3}}=2^{6}{\sqrt[{3}]{2^{2}}}\cdot 3^{3}{\sqrt[{3}]{3}}=1728{\sqrt[{3}]{12}}}$

 ← 基礎知識 微積分學 函數 → 代數