Algebra Cheat Sheet

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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

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}

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


0^{0}=\mathrm{Undefined} \frac{x}{0}=\mathrm{Undefined}
\log_{a}(b)=\mathrm{Undefined}\:,\: a\lt0 \log_{a}(b)=\mathrm{Undefined}\:,\: b\lt0
\log_{1}(a)=\mathrm{Undefined} \log_{0}(a)=\mathrm{Undefined}

Complex Number Rules

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

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