Álgebra
Regras de Número
\(a \cdot 0 = 0\)
\(1 \cdot a = a\)
Regras de Expansão
\(-(a \pm b) = -a \mp b\)
\(a(b + c) = ab + ac\)
\(a(b + c)(d + e) = abd + abe + acd + ace\)
\((a + b)(c + d) = ac + ad + bc + bd\)
\(-(-a) = a\)
Regras de Frações
\(\frac{a}{0} = 0, a \neq 0\)
\(\frac{1}{a} = a\)
\(\frac{a}{a} = 1\)
\(\frac{b}{a} = \frac{b}{a}\)
\(\frac{1}{\frac{a}{b}} = \frac{b}{a}\)
\(\frac{\frac{b}{a}}{\frac{1}{1}} = \frac{b}{a}\)
\(\frac{a}{\frac{1}{1}} = a\)
\(\frac{-b}{-a} = \frac{b}{a}\)
\(\frac{b}{-a} = -\frac{b}{a}\)
\(\frac{-b}{a} = -\frac{b}{a}\)
\(\frac{\frac{c}{b}}{a} = \frac{b}{a \cdot c}\)
\(\frac{a}{\frac{c}{b}} = \frac{c \cdot a}{b}\)
\(\frac{\frac{c}{b}}{1} = \frac{b}{c}\)
Regras de Absoluto
\(|-a| = |a|\)
\(|a| = a, a \geq 0\)
\(|ax| = |a||x|, a \geq 0\)
Regras de Expoente
\(1^a = 1\)
\(a^1 = a\)
\(a^0 = 1, a \neq 0\)
\(0^a = 0, a \neq 0\)
\((ab)^n = a^n b^n\)
\(\frac{a^m}{a^n} = a^{m-n}, m > n\)
\(\frac{a^m}{a^n} = \frac{1}{a^{n-m}}, n > m\)
\(a^{b+c} = a^b a^c\)
\((a^b)^c = a^{bc}\)
\((a^b)^c = a^{bc}\)
\(\left(\frac{b}{a}\right)^c = \frac{b^c}{a^c}\)
\((a \cdot b)^x = a^x \cdot b^x\)
Regras Radicais
\(\sqrt[c]{1} = 1\)
\(\sqrt[c]{0} = 0\)
\(\sqrt[1]{a} = a\)
\(\sqrt[n]{a^n} = a\)
\(\sqrt[c]{a} \sqrt[c]{a} = \sqrt[c]{a^2}\)
\(\sqrt[c]{a} = a^{1/c}, a \geq 0\)
\(\sqrt[n]{a} = |a|, n \text{ é par}\)
\(\sqrt[n]{a} = a, n \text{ é ímpar}\)
\(\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}, a,b \geq 0\)
\(\sqrt[n]{\frac{b}{a}} = \frac{\sqrt[n]{b}}{\sqrt[n]{a}}, a,b \geq 0\)
Regras de Fator
\(x^2 - y^2 = (x - y)(x + y)\)
\(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
\(x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \cdots + xy^{n-2} + y^{n-1})\)
\(x^n + y^n = (x + y)(x^{n-1} - x^{n-2}y + \cdots - xy^{n-2} + y^{n-1}), n \text{ é ímpar}\)
\(ax^2 - b = (\sqrt{a}x + \sqrt{b})(\sqrt{a}x - \sqrt{b})\)
\(ax^4 - b = (\sqrt[4]{a}x^2 + \sqrt[4]{b})(\sqrt[4]{a}x^2 - \sqrt[4]{b})\)
\(ax^{2n} - by^{2m} = (\sqrt[n]{a}x^n + \sqrt[m]{b}y^m)(\sqrt[n]{a}x^n - \sqrt[m]{b}y^m)\)
\(ax^4 - by^4 = (\sqrt[4]{a}x^2 + \sqrt[4]{b}y^2)(\sqrt[4]{a}x^2 - \sqrt[4]{b}y^2)\)
Regras de Fatorial
\(\frac{(n + m)!}{n!} = (n + 1) \cdot (n + 2) \cdots (n + m)\)
\(\frac{n!}{(n - m)!} = n \cdot (n - 1) \cdots (n - m + 1), n > m\)
\(0! = 1\)
\(n! = 1 \cdot 2 \cdots (n - 2) \cdot (n - 1) \cdot n\)
Regras de Log
\(\log_a 1 = 0\)
\(\log_a a = 1\)
\(\log_a(x^b) = b \cdot \log_a x\)
\(\log_a x = \frac{1}{\log_x a}\)
\(\log_a \frac{1}{x} = -\log_a x\)
\(\log_a(-x) = -\log_a x\)
\(\log_a b = \frac{\ln b}{\ln a}\)
\(\log_a(x^n) = n\log_a x\)