$$ \left.\begin{array}{l} a) \displaystyle{\frac{4}{\sqrt{2}}} & b) \displaystyle{\frac{3}{\sqrt[7]{3^2}}} & c) \displaystyle{\frac{4}{2-\sqrt{2}}} & d) \displaystyle{\frac{1}{2\sqrt{3}-\sqrt{5}}} \\ \end{array}\right. $$

$$ \left.\begin{array}{l} a) \displaystyle{2\sqrt{2}} & b) \displaystyle{\sqrt[7]{3^5}} & c) \displaystyle{4+2\sqrt{2}} & d) \displaystyle{\frac{2\sqrt{3}+\sqrt{5}}{7}} \\ \end{array}\right. $$

$$ \left.\begin{array}{l} a) \displaystyle{\sqrt{27}-\sqrt{50}+\sqrt{12}+\sqrt{8}} & b) \displaystyle{\sqrt{50a}-\sqrt{18a}} \\ c) \displaystyle{\sqrt{32}-\frac{\sqrt{50}}{2}+\frac{5}{\sqrt{18}}} & d) \displaystyle{5\sqrt{125}+6\sqrt{45}-7\sqrt{20}+\frac{3}{2}\sqrt{80}} \\ \end{array}\right. $$

$$ \left.\begin{array}{l} a) \displaystyle{-3\sqrt{2}+5\sqrt{3}} & b) \displaystyle{2\sqrt{2a}} \\ c) \displaystyle{\frac{7\sqrt{2}}{3}} & d) \displaystyle{35\sqrt{5}} \\ \end{array}\right. $$

Radicales:

$$ \left\{\begin{array}{l} \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. $$