Cálculos combinados | Potencias | Práctica 0 ejercicio 2

EJERCICIO 2:Calcule

(a) $\left(\dfrac{1}{4}-(\dfrac{2}{3}-\dfrac{1}{2})^2\right) ^{-2}$

(b) $\left(\left(4-\dfrac{1}{2}\right) ^2+\left(-3-\dfrac{1}{2}\right) ^2\right) ^{\frac{1}{2}}$


Solución:

(a) $\left(\dfrac{1}{4}-(\dfrac{2}{3}-\dfrac{1}{2})^2\right) ^{-2} = $

$\left(\dfrac{1}{4}-(\dfrac{4-3}{6})^2\right) ^{-2} = \left(\dfrac{1}{4}-(\dfrac{1}{6})^2\right) ^{-2} = $


$\left(\dfrac{1}{4}-(\dfrac{1^2}{6^2})\right) ^{-2} = \left(\dfrac{1}{4}-\dfrac{1}{36}\right) ^{-2} =$

$\left(\dfrac{9-1}{36}\right) ^{-2} =\left(\dfrac{\cancel{8}^2}{\cancel{36}_9}\right) ^{-2} = \left(\dfrac{2}{9}\right) ^{-2} = \left(\dfrac{9}{2}\right) ^{2} =$ $\boxed{\dfrac{81}{4}}$


(b) $\left(\left(4-\dfrac{1}{2}\right) ^2+\left(-3-\dfrac{1}{2}\right) ^2\right) ^{\frac{1}{2}} =$

$\left(\left(\dfrac{8-1}{2}\right) ^2+\left(\dfrac{-6-1}{2}\right) ^2\right) ^{\frac{1}{2}} =$

$\left(\left(\dfrac{7}{2}\right) ^2+\left(\dfrac{-7}{2}\right) ^2\right) ^{\frac{1}{2}} =$

$\left(\dfrac{7^2}{2^2}+\dfrac{(-7)^2}{2^2}\right) ^{\frac{1}{2}} =$

$\left(\dfrac{49}{4}+\dfrac{49}{4}\right) ^{\frac{1}{2}} =
\left(2\cdot \dfrac{49}{4}\right) ^{\frac{1}{2}} =\sqrt{ 2\cdot \dfrac{49}{4}} = $

$\sqrt{ 2}\cdot \sqrt{\dfrac{49}{4}} = \sqrt{ 2}\cdot \dfrac{\sqrt{49}}{\sqrt{4}} = \boxed{\dfrac{7}{2}\sqrt{ 2}} $