Algebra Cheat Sheet

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Algebra Cheat Sheet is a list of algebra formulas, a quick reference to learn Algebra.

Number Rules


\boxed{a\cdot 0=0}

\boxed{1\cdot a=a}

Expand Rules


\boxed{-\left(a\pm b\right)=-a\mp b}

\boxed{a\left(b+c\right)=ab+ac}

\boxed{a\left(b+c\right)\left(d+e\right)=abd+abe+acd+ace}

\boxed{\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd}

\boxed{-\left(-a\right)=a}

Fractions Rules


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

\boxed{\frac{a}{1}=a}

\boxed{\frac{a}{a}=1}

\boxed{\left(\frac{a}{b}\right)^{-1}=\frac{1}{\frac{a}{b}}=\frac{b}{a}}

\boxed{\left(\frac{a}{b}\right)^{-c}=\left(\left(\frac{a}{b}\right)^{-1}\right)^c=\left(\frac{b}{a}\right)^c}

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

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

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

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

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

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

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

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

Absolute Rules


\boxed{\left|-a\right|=\left|a\right|}

\boxed{\left|a\right|=a\:,\:a\ge 0}

\boxed{\left|ax\right|=a\left|x\right|\:,\:a\ge 0}

Exponent Rules


\boxed{1^a=1}

\boxed{a^1=a}

\boxed{a^0=1\:,\:a\ne 0}

\boxed{0^a=0\:,\:a\ne 0}

\boxed{\left(ab\right)^n=a^nb^n}

\boxed{\frac{a^m}{a^n}=a^{m-n}\:,\:m>n}

\boxed{\frac{a^m}{a^n}=\frac{1}{a^{n-m}}\:,\:n>m}

\boxed{a^{b+c}=a^ba^c}

\boxed{\left(a^b\right)^c=a^{b\cdot c}}

\boxed{a^{bx}=\left(a^b\right)^x}

\boxed{\left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}}

\boxed{a^c\cdot b^c=\left(a\cdot b\right)^c}

Radical Rules


\boxed{\sqrt{1}=1}

\boxed{\sqrt{0}=0}

\boxed{\sqrt[n]{a}=a^{\frac{1}{n}}}

\boxed{\sqrt[n]{a^m}=a^{\frac{m}{n}}}

\boxed{\sqrt{a}\sqrt{a}=a}

\boxed{\sqrt[n]{a^n}=a,\:a\ge 0}

\boxed{\sqrt[n]{a^n}=|a|,\:\mathrm{n\:is\:even}}

\boxed{\sqrt[n]{a^n}=a,\:\mathrm{n\:is\:odd}}

\boxed{\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:a,b\ge 0}

\boxed{\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},\:a,b\ge 0}

Factor Rules


\boxed{x^2-y^2=\left(x-y\right)\left(x+y\right)}

\boxed{x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)}

\boxed{x^n-y^n=\left(x-y\right)\left(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1}\right)}

\boxed{x^n+y^n=\left(x+y\right)\left(x^{n-1}-x^{n-2}y+\dots -xy^{n-2}+y^{n-1}\right)\quad \quad \mathrm{n\:is\:odd}}

\boxed{ax^{\left(2n\right)}-b=\left(\sqrt{a}x^n+\sqrt{b}\right)\left(\sqrt{a}x^n-\sqrt{b}\right)}

\boxed{ax^{\left(4\right)}-b=\left(\sqrt{a}x^2+\sqrt{b}\right)\left(\sqrt{a}x^2-\sqrt{b}\right)}

\boxed{ax^{\left(2n\right)}-by^{\left(2m\right)}=\left(\sqrt{a}x^n+\sqrt{b}y^m\right)\left(\sqrt{a}x^n-\sqrt{b}y^m\right)}

\boxed{ax^{\left(4\right)}-by^{\left(4\right)}=\left(\sqrt{a}x^2+\sqrt{b}y^2\right)\left(\sqrt{a}x^2-\sqrt{b}y^2\right)}

Factorial Rules


\boxed{\frac{n!}{\left(n+m\right)!}=\frac{1}{\left(n+1\right)\cdot \left(n+2\right)\cdots \left(n+m\right)}}

\boxed{\frac{n!}{\left(n-m\right)!}=n\cdot \left(n-1\right)\cdots \left(n-m+1\right),n>m}

\boxed{0!=1}

\boxed{n!=1\cdot 2\cdots \left(n-2\right)\cdot \left(n-1\right)\cdot n}

Log Rules


\boxed{\log \left(1\right)=0}

\boxed{\log _a\left(a\right)=1}

\boxed{\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)}

\boxed{\log _{a^b}\left(x\right)=\frac{1}{b}\log _a\left(x\right)}

\boxed{\log _a\left(\frac{1}{x}\right)=-\log _a\left(x\right)}

\boxed{\log _{\frac{1}{a}}\left(x\right)=-\log _a\left(x\right)}

\boxed{\log _a\left(b\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}}

\boxed{\log _x\left(x^n\right)=n}

\boxed{\log _x\left(\left(\frac{1}{x}\right)^n\right)=-n}

\boxed{a^{\log _a\left(b\right)}=b}

Undefined


\boxed{0^0=\mathrm{Undefined}}


\boxed{\frac{x}{0}=\mathrm{Undefined}}


\boxed{\log _a\left(b\right)=\mathrm{Undefined}\:,\:a\le 0}


\boxed{\log _a\left(b\right)=\mathrm{Undefined}\:,\:b\le 0}


\boxed{\log _1\left(a\right)=\mathrm{Undefined}}

Complex Number Rules


\boxed{i^2=-1}

\boxed{\sqrt{-1}=i}

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

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