# Algebra

## Algebra Formulas

$-(-a)=a$

$-(a + b)= -a - b$

$-(a - b)= -a + b$

$a(b+c)=ab+ac$

$a(b+c)(d+e)=abd+abe+acd+ace$

$(a+b)(c+d)=ac+ad+bc+bd$

### Factor Rules

$a^{2}-b^{2} = (a-b)(a+b)$

$a^{2}+b^{2} =$ - there is no such a formula.

$a^{3}-b^{3} = (a-b)(a^{2}+ab+ b^{2})$

$a^{3}+b^{3} = (a+b)(a^{2}-ab+ b^{2})$

### Fraction Rules

$\frac{0}{a}=0$     a ≠ 0

$\frac{a}{0}=$ the operation is undefined. Dividing by zero is not allowed.

$\frac{a}{1}=a$

$\frac{a}{a}=1$

$(\frac{a}{b})^{-1}=\frac{1}{\frac{a}{b}}=\frac{b}{a}$

$(\frac{a}{b})^{-c}=((\frac{a}{b})^{-1})^{c}=(\frac{b}{a})^{c}$

$a^{-1}=\frac{1}{a}$

$a^{-b}=\frac{1}{a^b}$

$\frac{-a}{-b}=\frac{a}{b}$

$\frac{-a}{b}=-\frac{a}{b}$

$\frac{a}{-b}=-\frac{a}{b}$

$\frac{a}{\frac{b}{c}}=\frac{a\cdot c}{b}$

$\frac{\frac{b}{c}}{a}=\frac{b}{c \cdot a}$

$\frac{1}{\frac{b}{c}}=\frac{c}{b}$

### Modules

|x|=x     if x ≥ 0

|x|=-x     if x < 0

### Exponent Rules

$1^{a}=1$

$a^{1}=a$

$a^{0}=1$     if a ≠ 0

$0^{a}=0$     if a ≠ 0

$(ab)^n=a^nb^n$

$\frac{a^m}{a^n}=a^{m-n}$     if m > n

$\frac{a^m}{a^n}=\frac{1}{a^{n-m}}$     if m < n

$a^{b+c}=a^{b}a^{c}$

$(a^{b})^{c}=a^{b\cdot c}$

$a^{bx}=(a^b)^x$

$(\frac{a}{b})^{c}=\frac{a^{c}}{b^{c}}$

$a^c \cdot b^c=(a\cdot b)^{c}$

$a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}$

$\sqrt{-a}=\sqrt{-1}\sqrt{a}$

$\sqrt[n]{a\cdot b}=\sqrt[n]{a}\sqrt[n]{b}$

### Factorial

$n!=1 \cdot 2 \cdots(n-2) \cdot (n-1) \cdot n$

$0!=1$

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