Potencias

$$ \left\{\begin{array}{lll} & \underbrace{a \cdot a \cdots a \cdot a}_{\hbox{n veces}} = a^n & \\ & & \\ a^0=1 & a^1=a & a^n \cdot a^m = a^{n+m}\\ \frac{a^n}{a^m} = a^{n-m} & (a^n)^m = a^{n \cdot m} & a^{-n} = \frac{1}{a^n} \\ (a \cdot b)^{n} = a^n \cdot b^n & \left( \frac{a}{b} \right)^{n} = \frac{a^n}{b^n} & \left( \frac{a}{b} \right)^{-n} = \left( \frac{b}{a} \right)^{n} = \frac{b^n}{a^n} \\ \end{array}\right. $$

Radicales

$$ \left\{\begin{array}{lll} \mathbf{\hbox{(Propiedades)}} \left\{\begin{array}{l} \sqrt[n]{a^m} = a^{\frac{m}{n}} & \frac{1}{\sqrt[n]{a^m}} = a^{-\frac{m}{n}} & \sqrt[n]{a} = a^{\frac{1}{n}} & \frac{1}{\sqrt[n]{a}} = a^{-\frac{1}{n}} \\ \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} & \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} & \sqrt[m]{\sqrt[n]{a}} = \sqrt[n \cdot m]{a} & \sqrt[m]{b\sqrt[n]{a}} = \sqrt[n \cdot m]{b^n a}\\ \left( \sqrt[n]{a} \right)^{m} = \sqrt[n]{a^m} & \sqrt[n]{a^n} = a & \sqrt[n \cdot p]{a^p} = \sqrt[n]{a} & a\sqrt[n]{b} = \sqrt[n]{a^n b} \\ \end{array}\right.\\ \\ \mathbf{\hbox{(Radicales con distinto índice)}} \left\{\begin{array}{l} \sqrt[n]{a} \cdot \sqrt[m]{b} = \sqrt[p]{a^{\frac{p}{n}} \cdot b^{\frac{p}{m}}}\\ \frac{\sqrt[n]{a}}{\sqrt[m]{b}} = \sqrt[p]{\frac{a^{\frac{p}{n}}}{b^{\frac{p}{m}}}}\\ \end{array}\right., p = m.c.m.(n,m) \\ \\ \mathbf{\hbox{(Suma y resta de radicales equivalentes)}} \left\{\begin{array}{l} a\sqrt[n]{c} + b\sqrt[n]{c} = (a + b)\sqrt[n]{c}\\ a\sqrt[n]{c} - b\sqrt[n]{c} = (a - b)\sqrt[n]{c}\\ \end{array}\right.\\ \\ \mathbf{\hbox{(Racionalizar)}} \left\{\begin{array}{l} \frac{a}{\sqrt{b}}=\frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a \sqrt{b}}{\sqrt{b^2}} = \frac{a \sqrt{b}}{b} \\ \frac{a}{\sqrt[n]{b^m}} = \frac{a}{\sqrt[n]{b^m}} \cdot \frac{\sqrt[n]{b^{n-m}}}{\sqrt[n]{b^{n-m}}} = \frac{a\sqrt[n]{b^{n-m}}}{\sqrt[n]{b^n}} = \frac{a\sqrt[n]{b^{n-m}}}{b} \\ \frac{a}{b+\sqrt{c}} = \frac{a}{b+\sqrt{c}} \cdot \frac{b-\sqrt{c}}{b-\sqrt{c}} = \frac{a(b-\sqrt{c})}{b^2-\sqrt{c^2}} = \frac{ab-a\sqrt{c}}{b^2-c} \\ \frac{a}{b-\sqrt{c}} = \frac{a}{b-\sqrt{c}} \cdot \frac{b+\sqrt{c}}{b+\sqrt{c}} = \frac{a(b+\sqrt{c})}{b^2-\sqrt{c^2}} = \frac{ab+a\sqrt{c}}{b^2-c} \\ \frac{a}{\sqrt{b}+\sqrt{c}} = \frac{a}{\sqrt{b}+\sqrt{c}} \cdot \frac{\sqrt{b}-\sqrt{c}}{\sqrt{b}-\sqrt{c}} = \frac{a(\sqrt{b}-\sqrt{c})}{\sqrt{b^2}-\sqrt{c^2}} = \frac{a\sqrt{b}-a\sqrt{c}}{b-c} \\ \frac{a}{\sqrt{b}-\sqrt{c}} = \frac{a}{\sqrt{b}-\sqrt{c}} \cdot \frac{\sqrt{b}+\sqrt{c}}{\sqrt{b}+\sqrt{c}} = \frac{a(\sqrt{b}+\sqrt{c})}{\sqrt{b^2}-\sqrt{c^2}} = \frac{a\sqrt{b}+a\sqrt{c}}{b-c} \\ \end{array}\right.\\ \end{array}\right. $$

Logaritmos

$$ \left\{\begin{array}{c} \hbox{Dados } a,b>0, a \neq 1, \hbox{ } \log_{a}{b} = x \Leftrightarrow a^x = b\\ \\ \mathbf{\hbox{(Propiedades)}} \left\{\begin{array}{ll} \log_{a}{1} = 0 \hbox{ , } \log_{a}{a} = 1 & \log_{a}{a^n} = n \\ \log_{a}{(x \cdot y)} = \log_{a}{x} + \log_{a}{y} & \log_{a}{\left( \frac{x}{y} \right)} = \log_{a}{x} - \log_{a}{y} \\ \log_{a}{x^n} = n \log_{a}{x} & \log_{a}{\sqrt[n]{x}} = \frac{1}{n} \log_{a}{x} \\ \end{array}\right.\\ \\ \mathbf{\hbox{(Notación)}} \left\{\begin{array}{l} \log_{10}{x} = \log{x} \hbox{ (Logaritmo decimal)} \\ \log_{e}{x} = \ln{x} \hbox{ (Logaritmo neperiano)} \\ \end{array}\right.\\ \mathbf{\hbox{(Cambio de base)}} \left.\begin{array}{l} \\ \log_{a}{x} = \frac{\log_{b}{x}}{\log_{b}{a}} = \frac{\log{x}}{\log{a}} = \frac{\ln{x}}{\ln{a}} \\ \\ \end{array}\right.\\ \mathbf{\hbox{(Exponencial y Logaritmo)}} \left.\begin{array}{ll} \hbox{Dados } a,b>0, a,b \neq 1, & a^x = e^{x \cdot \ln(a)} = b^{x \cdot \log_{b}{a}} \\ \end{array}\right.\\ \end{array}\right. $$

Radicales y Logaritmos (Clica para mostrar)