Question

Racionalização de frações$\\ \\ \frac{21}{ \sqrt{3+ \sqrt{7} } }$,
tenho duvida principalmente na multiplicação das bases

Answer 1

$Fra\c{c}\~ao:\\\\ \dfrac{21}{\sqrt{3+\sqrt{7}}}\\\\\\ Racionalizando:\\\\ \dfrac{21}{\sqrt{3+\sqrt{7}}}\times \dfrac{\sqrt{3-\sqrt{7}}}{\sqrt{3-\sqrt{7}}}\\\\\\ =\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{(3+\sqrt{7}).(3-\sqrt{7})}}\\\\\\ =\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{(9-3\sqrt{7}+3\sqrt{7}-7)}}\\\\\\ =\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{2}}$


$Racionalizando\ novamente:\\\\ =\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}}\\\\\\ =\dfrac{21\sqrt{(3-\sqrt{7})\times 2}}{2}\\\\\\ \boxed{=\dfrac{21\sqrt{6-2\sqrt{7}}}{2}}$


Esperto ter ajudado.
Bons estudos!