Number Rules
| $a\cdot0=0$a·0=0 | $1\cdot a=a$1·a=a |
Expand Rules
| $-\left(a\pm b\right)=-a\mp b$−(a±b)=−a∓b | $a\left(b+c\right)=ab+ac$a(b+c)=ab+ac |
| $a\left(b+c\right)\left(d+e\right)=abd+abe+acd+ace$a(b+c)(d+e)=abd+abe+acd+ace | $\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$(a+b)(c+d)=ac+ad+bc+bd |
| $-\left(-a\right)=a$−(−a)=a | $$ |
Fractions Rules
| $\frac{0}{a}=0\:,\:a\ne0$0a =0 , a≠0 | $\frac{a}{1}=a$a1 =a |
| $\frac{a}{a}=1$aa =1 | $\left(\frac{a}{b}\right)^{-1}=\frac{1}{\frac{a}{b}}=\frac{b}{a}$(ab )−1=1ab =ba |
| $\left(\frac{a}{b}\right)^{-c}=\left(\left(\frac{a}{b}\right)^{-1}\right)^c=\left(\frac{b}{a}\right)^c$(ab )−c=((ab )−1)c=(ba )c | $a^{-1}=\frac{1}{a}$a−1=1a |
| $a^{-b}=\frac{1}{a^b}$a−b=1ab | $\frac{-a}{-b}=\frac{a}{b}$−a−b =ab |
| $\frac{-a}{b}=-\frac{a}{b}$−ab =−ab | $\frac{a}{-b}=-\frac{a}{b}$a−b =−ab |
| $\frac{a}{\frac{b}{c}}=\frac{a\cdot c}{b}$abc =a·cb | $\frac{\frac{b}{c}}{a}=\frac{b}{c\cdot a}$bc a =bc·a |
| $\frac{1}{\frac{b}{c}}=\frac{c}{b}$1bc =cb | $$ |
Absolute Rules
| $\left|-a\right|=a$|−a|=a | $\left|a\right|=a\:,\:a\ge0$|a|=a , a≥0 |
| $\left|-a\right|=\left|a\right|$|−a|=|a| | $\left|ax\right|=a\left|x\right|\:,\:a\ge0$|ax|=a|x| , a≥0 |
Exponent Rules
| $1^a=1$1a=1 | $a^1=a$a1=a |
| $a^0=1\:,\:a\ne0$a0=1 , a≠0 | $0^a=0\:,\:a\ne0$0a=0 , a≠0 |
| $\left(ab\right)^n=a^nb^n$(ab)n=anbn | $\frac{a^m}{a^n}=a^{m-n}\:,\:m>n$aman =am−n , m>n |
| $\frac{a^m}{a^n}=\frac{1}{a^{n-m}}\:,\:n>m$aman =1an−m , n>m | $a^{b+c}=a^ba^c$ab+c=abac |
| $\left(a^b\right)^c=a^{b\cdot c}$(ab)c=ab·c | $a^{bx}=\left(a^b\right)^x$abx=(ab)x |
| $\left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$(ab )c=acbc | $a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m$amn =(n√a)m |
| $a^c\cdot b^c=\left(a\cdot b\right)^c$ac·bc=(a·b)c | $\sqrt[n]{a\cdot b}=\sqrt[n]{a}\sqrt[n]{b}$n√a·b=n√an√b |
Factor Rules
| $x^2-y^2=\left(x-y\right)\left(x+y\right)$x2−y2=(x−y)(x+y) |
| $x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)$x3+y3=(x+y)(x2−xy+y2) |
| $x^n-y^n=\left(x-y\right)\left(x^{n-1}+x^{n-2}y+\dots+xy^{n-2}+y^{n-1}\right)$xn−yn=(x−y)(xn−1+xn−2y+…+xyn−2+yn−1) |
| $x^n+y^n=\left(x+y\right)\left(x^{n-1}-x^{n-2}y+\dots-xy^{n-2}+y^{n-1}\right)\quad\quad\mathrm{n\:is\:odd}$xn+yn=(x+y)(xn−1−xn−2y+…−xyn−2+yn−1) n is odd |
| $ax^{\left(2n\right)}-b=\left(\sqrt{a}x^n+\sqrt{b}\right)\left(\sqrt{a}x^n-\sqrt{b}\right)$ax(2n)−b=(√axn+√b)(√axn−√b) |
| $ax^{\left(4\right)}-b=\left(\sqrt{a}x^2+\sqrt{b}\right)\left(\sqrt{a}x^2-\sqrt{b}\right)$ax(4)−b=(√ax2+√b)(√ax2−√b) |
| $ax^{\left(2n\right)}-by^{\left(2m\right)}=\left(\sqrt{a}x^n+\sqrt{b}y^m\right)\left(\sqrt{a}x^n-\sqrt{b}y^m\right)$ax(2n)−by(2m)=(√axn+√bym)(√axn−√bym) |
| $ax^{\left(4\right)}-by^{\left(4\right)}=\left(\sqrt{a}x^2+\sqrt{b}y^2\right)\left(\sqrt{a}x^2-\sqrt{b}y^2\right)$ax(4)−by(4)=(√ax2+√by2)(√ax2−√by2) |
Factorial Rules
| $\frac{n!}{\left(n+m\right)!}=\frac{1}{\left(n+1\right)\cdot\left(n+2\right)\cdots\left(n+m\right)}$n!(n+m)! =1(n+1)·(n+2)⋯(n+m) | $\frac{n!}{\left(n-m\right)!}=n\cdot\left(n-1\right)\cdots\left(n-m+1\right),n>m$n!(n−m)! =n·(n−1)⋯(n−m+1),n>m |
| $0!=1$0!=1 | $n!=1\cdot2\cdots\left(n-2\right)\cdot\left(n-1\right)\cdot n$n!=1·2⋯(n−2)·(n−1)·n |
Log Rules
| $\log\left(0\right)=-\infty$log(0)=−∞ | $\log\left(1\right)=0$log(1)=0 |
| $\log_a\left(a\right)=1$loga(a)=1 | $\log_a\left(x^b\right)=b\cdot\log_a\left(x\right)$loga(xb)=b·loga(x) |
| $\log_{a^b}\left(x\right)=\frac{1}{b}\log_a\left(x\right)$logab(x)=1b loga(x) | $\log_a\left(\frac{1}{x}\right)=-\log_a\left(x\right)$loga(1x )=−loga(x) |
| $\log_{\frac{1}{a}}\left(x\right)=-\log_a\left(x\right)$log1a (x)=−loga(x) | $\log_{x^n}\left(x\right)=\frac{1}{n}$logxn(x)=1n |
| $\log_a\left(b\right)=\frac{\ln\left(b\right)}{\ln\left(a\right)}$loga(b)=ln(b)ln(a) | $\log_x\left(x^n\right)=n$logx(xn)=n |
| $\log_x\left(\left(\frac{1}{x}\right)^n\right)=-n$logx((1x )n)=−n | $a^{\log_a\left(b\right)}=b$aloga(b)=b |
Undefined
| $0^0=\mathrm{Undefined}$00=Undefined | $\frac{x}{0}=\mathrm{Undefined}$x0 =Undefined |
| $\log_a\left(b\right)=\mathrm{Undefined}\:,\:a\le0$loga(b)=Undefined , a≤0 | $\log_a\left(b\right)=\mathrm{Undefined}\:,\:b\le0$loga(b)=Undefined , b≤0 |
| $\log_1\left(a\right)=\mathrm{Undefined}$log1(a)=Undefined |
Complex Number Rules
| $i^2=-1$i2=−1 | $\sqrt{-1}=i$√−1=i |
| $\sqrt{-a}=\sqrt{-1}\sqrt{a}$√−a=√−1√a | $$ |
