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)
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