Question

$\frac{1}{ \sqrt{6 + \sqrt{5} } }$
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Answer 1

$\displaystyle \frac{1}{\sqrt{6+\sqrt5}} \\\\\\ \underline{\text{racionalizando}}: \\\\ \frac{1}{(\sqrt{6+\sqrt5})}.\frac{\sqrt{6+\sqrt5}}{(\sqrt{6+\sqrt5})} \\\\\\ \frac{\sqrt{6+\sqrt5}}{(\sqrt{6+\sqrt5})^2} \\\\\\ \frac{\sqrt{6+\sqrt5}}{6+\sqrt5} \\\\ \underline{\text {racionalizando novamente}} : \\\\ \frac{\sqrt{6+\sqrt5}}{(6+\sqrt5)}.\frac{6-\sqr5}{(6-\sqrt5)} \\\\\\ \frac{(6-\sqrt5).\sqrt{6+\sqrt5}}{6^2-(\sqrt5)^2} \\\\ \frac{(6-\sqrt5).\sqrt{6+\sqrt5}}{36-5} \to \frac{(6-\sqrt5).\sqrt{6+\sqrt5}}{31}$

Fazendo a distributiva, temos :

$\displaystyle \frac{6\sqrt{6+\sqrt5}-\sqrt5.\sqrt{6+\sqrt5}}{31} \\\\\\ \frac{6\sqrt{6+\sqrt5} - \sqrt{5(6+\sqrt5)}}{31} \\\\\\ \frac{6\sqrt{6+\sqrt5} -\sqrt{30+5\sqrt{5}}}{31}$

Portanto, temos :

$\huge\boxed{\frac{(6-\sqrt5).\sqrt{6+\sqrt5}}{31}\ }\checkmark \\\\\\\\ \text{ou} \\\\\\\huge\boxed{\displaystyle \frac{6\sqrt{6+\sqrt5} -\sqrt{30+5\sqrt{5}}}{31}} \checkmark$