Ejercicios resueltos - Radicación -

Efectuar aplicando las propiedades de la radicación y hasta llegar a la mínima expresión

a)  $\displaystyle\frac{a^{-2/3}\sqrt[4]{b^5}}{\sqrt[3]{a}b^{-6/5}}$→$\displaystyle\frac{\sqrt[4]{b^5}b^{6/5}}{\sqrt[3]aa^{2/3}}$→$\displaystyle\frac{\sqrt[4]{b^5}\sqrt[5]{b^6}}{\sqrt[3]{a}\sqrt[3]{a^2}}$→$\displaystyle\frac{\sqrt[20]{(b^5)^5(b^6)^4}}{\sqrt[3]{aa^2}}$→$\displaystyle\frac{\sqrt[20]{b^{25}b^{24}}}{a}$→$\displaystyle\frac{b^2\sqrt[20]{b^9}}{a}\bullet$

b)  $\displaystyle\frac{(\sqrt[3]{xy^2})^4\sqrt[3]{16x\sqrt{y}}}{2y^2\displaystyle\sqrt[6]{\frac{x^4}{y}}}$→$\displaystyle\frac{\sqrt[3]{x^4y^8}\sqrt[3]{\sqrt{2^8x^2y}}}{2y^2\displaystyle\sqrt[6]{\frac{x^4}{y}}}$→$\displaystyle\frac{\sqrt[3]{x^4y^8}\sqrt[6]{2^8x^2y}}{2y^2\displaystyle\sqrt[6]{\frac{x^4}{y}}}$→$\displaystyle\frac{\sqrt[6]{x^8y^{16}2^8x^2y}}{2y^2\displaystyle\sqrt[6]{\frac{x^4}{y}}}$→$\displaystyle\frac{\color{red}{2y^2}x\sqrt[6]{x^4y^52^2}}{\color{red}{2y^2}\displaystyle\sqrt[6]{\frac{x^4}{y}}}$→$x\displaystyle\sqrt{\frac{x^4y^52^2}{\frac{x^4}{y}}}$→$x\displaystyle\sqrt{\frac{\color{red}{x^4}y^62^2}{\color{red}{x^4}}}$→$xy\displaystyle\sqrt[3]{2}\bullet$

c)  $\displaystyle\frac{\sqrt{x^2-1}}{\sqrt{x^2+2x+1}}$→$\displaystyle\sqrt{\frac{x^2-1}{x^2+2x+1}}$→$\displaystyle\sqrt{\frac{\color{red}{(x+1)}(x-1)}{\color{red}{(x+1)}(x+1)}}$→$\displaystyle\sqrt{\frac{(x-1)}{(x+1)}}\bullet$

d)  $\displaystyle\frac{\sqrt{xy^3}\sqrt{2x^2y}}{\sqrt{6x^3y^4}}$→$\displaystyle\frac{\sqrt{xy^32x^2y}}{\sqrt{6x^3y^4}}$→$\sqrt{\displaystyle\frac{\color{red}{x^3y^42}}{3.\color{red}{2x^3y^4}}}$→$\displaystyle\sqrt{\displaystyle\frac{1}{3}}\bullet$

e)  $3\sqrt{\sqrt{2}}-\displaystyle\frac{1}{2}\sqrt[4]{2}+\sqrt[12]{2^3}$→$3\color{red}{\sqrt[4]{2}}-\displaystyle\frac{1}{2}\color{red}{\sqrt[4]{2}}+\color{red}{\sqrt[4]{2}}$→$\displaystyle\frac{7}{2}\sqrt[4]{2}\bullet$