Question

A fração $\frac{2. (\sqrt{6}+ \sqrt{2}) }{3. \sqrt{2+ \sqrt{3} } }$ é igual a:
a) $\frac{2 \sqrt{2} }{3}$
b) 1
c) $\frac{4}{3}$
d) $\frac{16}{9}$
e) $\frac{4}{9}$

Answer 1

$\displaystyle E=\frac{2(\sqrt{6}+ \sqrt{2}) }{3 \sqrt{2+ \sqrt{3} } } \\ \\ \\ E=\frac{2(\sqrt{6}+ \sqrt{2})}{3\sqrt{2+\sqrt{3}}} \cdot\frac{\sqrt{2-\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\\ \\ \\ E=\dfrac{2(\sqrt{6}+ \sqrt{2})(\sqrt{2-\sqrt{3}})}{3}\\ \\ \\ E=\dfrac{(\sqrt{6}+ \sqrt{2})(\sqrt{8-2\sqrt{12}})}{3}\\ \\ \\ E=\dfrac{(\sqrt{6}+ \sqrt{2})(\sqrt{6}-\sqrt{2})}{3}\\ \\ \\ \boxed{E=\frac{4}{3}}$