Question

Como Racionalizar Passo a Passo √3+3 / 3-√3

Answer 1

$\dfrac{3+\sqrt{3}}{3-\sqrt{3}}$


Multiplicando e dividindo pelo conjugado do denominador, que é $\big(3+\sqrt{3}\big):$

$=\dfrac{3+\sqrt{3}}{3-\sqrt{3}}\cdot \dfrac{3+\sqrt{3}}{3+\sqrt{3}}\\\\\\ =\dfrac{\big(3+\sqrt{3}\big)\cdot \big(3+\sqrt{3}\big)}{\big(3-\sqrt{3}\big)\cdot \big(3+\sqrt{3}\big)}\\\\\\ =\dfrac{\big(3+\sqrt{3}\big)^2}{3^2-\big(\sqrt{3}\big)^2}\\\\\\ =\dfrac{3^2+2\cdot 3\cdot \sqrt{3}+\big(\sqrt{3}\big)^2}{9-3}\\\\\\ =\dfrac{9+6\sqrt{3}+3}{6}\\\\\\ =\dfrac{12+6\sqrt{3}}{6}\\\\\\ =\dfrac{\diagup\!\!\!\! 6\cdot \big(2+\sqrt{3}\big)}{\diagup\!\!\!\! 6}\\\\\\ =\boxed{\begin{array}{c}2+\sqrt{3} \end{array}}$


Bons estudos! :-)