# Algebra Cheat Sheet

• Algebra
• Number Rules
• Expand Rules
• Fractions Rules
• Absolute Rules
• Exponent Rules
• Factor Rules
• Factorial Rules
• Log Rules
• Undefined
• Complex Number Rules
• Trigonometry
• Basic Identities
• Pythagorean Identities
• Double Angle Identities
• Sum Difference Identities
• Product To Sum Identities
• Triple Angle Identities
• Function Ranges
• Function Values
• Limits
• Limit Properties
• Limit to Infinity Properties
• Indeterminate Forms
• Common Limits
• Limit Rules
• Derivatives
• Derivatives Rules
• Common Derivatives
• Trigonometric Derivatives
• Arc Trigonometric Derivatives
• Hyperbolic Derivatives
• Arc Hyperbolic Derivatives
• Integrals
• Common Integrals
• Trigonometric Integrals
• Arc Trigonometric Integrals
• Hyperbolic Integrals
• Integrals of Special Functions
• Indefinite Integrals Rules
• Definite Integrals Rules

## Number Rules

 a\cdot 0=0 1\cdot a=a

## Expand Rules

 -(a\pm b)=-a\mp b a(b+c)=ab+ac a(b+c)(d+e)=abd+abe+acd+ace (a+b)(c+d)=ac+ad+bc+bd -(-a)=a

## Fractions Rules

 \frac{0}{a}=0 \: ,\: a\ne 0 \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}

## Absolute Rules

 \left|-a\right|=a \left|a\right|=a \: ,\: a\ge0 \left| -a \right| = \left| a \right| \left| ax\right| = a \left| x\right| \: , \: a\ge 0

## Exponent Rules

 1^{a}=1 a^{1}=a a^{0}=1\:,\: a\ne 0 0^{a}=0\:,\: a\ne 0 (ab)^n=a^{n}b^{n} \frac{a^m}{a^n}=a^{m-n}\:,\: m>n \frac{a^m}{a^n}=\frac{1}{a^{n-m}}\:,\: n>m 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^{\frac{m}{n}}=(\sqrt[n]{a})^{m} a^c \cdot b^c=(a\cdot b)^{c} \sqrt[n]{a\cdot b}=\sqrt[n]{a}\sqrt[n]{b}

## Factor Rules

 x^{2}-y^{2} = (x-y)(x+y) x^{3}+y^{3} = (x+y)(x^{2}-xy+ y^{2}) x^{n}-y^{n} = (x-y)(x^{n-1}+x^{n-2}y+ \dots + xy^{n-2} + y^{n-1}) x^{n}+y^{n} = (x+y)(x^{n-1}-x^{n-2}y+ \dots - xy^{n-2} + y^{n-1}) \quad \quad \mathrm{n \: is \: odd} ax^(2n)-b = (\sqrt{a}x^n+\sqrt{b})(\sqrt{a}x^n-\sqrt{b}) ax^(4)-b = (\sqrt{a}x^2+\sqrt{b})(\sqrt{a}x^2-\sqrt{b}) ax^(2n)-by^(2m) = (\sqrt{a}x^n+\sqrt{b}y^m)(\sqrt{a}x^n-\sqrt{b}y^m) ax^(4)-by^(4) = (\sqrt{a}x^2+\sqrt{b}y^2)(\sqrt{a}x^2-\sqrt{b}y^2)

## Factorial Rules

 \frac{n!}{(n+m)!}=\frac{1}{(n+1)\cdot(n+2)\cdots(n+m)} \frac{n!}{(n-m)!}=n\cdot(n-1)\cdots(n-m+1), n>m 0!=1 n!=1\cdot2\cdots(n-2)\cdot(n-1)\cdot n

## Log Rules

 \log(0)=-\infty \log(1)=0 \log_a(a)=1 \log_{a}(x^b)=b\cdot\log_{a}(x) \log_{a^b}(x)=\frac{1}{b}\log_{a}(x) \log_{a}(\frac{1}{x})=-\log_{a}(x) \log_{\frac{1}{a}}(x)=-\log_{a}(x) \log_{x^n}(x)=\frac{1}{n} \log_{a}(b)=\frac{\ln(b)}{\ln(a)} \log_{x}(x^n)=n \log_{x}((\frac{1}{x})^{n})=-n a^{\log_{a}(b)}=b

## Undefined

 0^{0}=\mathrm{Undefined} \frac{x}{0}=\mathrm{Undefined} \log_{a}(b)=\mathrm{Undefined}\:,\: a\le0 \log_{a}(b)=\mathrm{Undefined}\:,\: b\le0 \log_{1}(a)=\mathrm{Undefined}

## Complex Number Rules

 i^{2}=-1 \sqrt{-1}=i \sqrt{-a}=\sqrt{-1}\sqrt{a}

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