Ejercicios Resueltos - Potenciación

Efectuar, aplicando las propiedades de la potenciación, hasta llegar a la mínima expresión.

a)  $\displaystyle\frac{2^4\left[2(2ab^{-1})^{-2}(-a)^{-3}\right]^2}{((a)^{-2}(-b))^3}$ → $\displaystyle\frac{2^4(2ab^{-1})^{-4}(-a)^{-6}}{(a)^{-6}(-b)^3}$→ $-\displaystyle\frac{\color{red}{2^42^{-4}}a^{-4}\color{red}{b^3}b\color{red}{a^{-6}}}{\color{red}{a^{-6}}\color{red}{b^3}}$→ $-\displaystyle\frac{b}{a^4}\bullet$

b)  $\left(\displaystyle\frac{(xy^2 z^3)^{-2}(x^2yz^2)^3}{(x^3y^2x^4)^2(x+y)^0}\right)^{-1}$→$\left(\displaystyle\frac{(x^{-2}y^{-4}\color{red}{z^{-6}})(x^6y^3\color{red}{z^6})}{(x^6y^4x^8)(1)}\right)^{-1}$→$\left(\displaystyle\frac{x^4y^{-1}}{x^6y^4x^8}\right)^{-1}$→$\left(\displaystyle\frac{1}{x^{10}y^5}\right)^{-1}$→$x^{10}y^5\bullet$

c)  $\displaystyle\frac{(-3\sqrt{2})(-3)^3(100)^0}{81(\sqrt{2})^3}$→$\displaystyle\frac{\color{red}{(-3\sqrt{2}})(\color{red}{-3^3})}{\color{red}{3^4}(\sqrt{2})^2(\color{red}{\sqrt{2}})}$→$\displaystyle\frac{1}{2}\bullet$

d)  $\displaystyle\frac{(-a^3b^{-3})^2+(2a^2b^{-2})-\left[a^2(-b)^{-2}\right]^3}{(-a^2b^3)^{-2}}$→$\displaystyle\frac{a^6b^{-6}+2a^2b^{-2}-a^6(-b)^{-6}}{-a^{-4}b^{-6}}$→$\displaystyle\frac{\color{red}{a^6b^{-6}}+2a^2b^{-2}\color{red}{-a^6b^{-6}}}{-a^{-4}b^{-6}}$→$-2a^6b^4\bullet$