Question

Racionalize a expressão:

$\frac{21}{\sqrt{3+\sqrt{7} }}$

Answer 1

$\dfrac{21}{\sqrt{3+\sqrt{7}}}~=\\\\\\\\=~\dfrac{21}{\sqrt{3+\sqrt{7}}}~\cdot~\dfrac{\sqrt{3-\sqrt{7}}}{\sqrt{3-\sqrt{7}}}\\\\\\\\=~\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{(3+\sqrt{7}\,)\cdot(3-\sqrt{7}\,)}}$

$=~\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{3\,.\,3~+~3\,.\,\sqrt{7}~-~\sqrt{7}\,.\,3~-~\sqrt{7}\,.\,\sqrt{7}}}\\\\\\\\=~\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{9~+~3\sqrt{7}~-~3\sqrt{7}~-~7}}\\\\\\\\=~\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{9~-~7}}$

$=~\dfrac{21\sqrt{3-\sqrt{7}}}{\sqrt{2}}~\cdot~\dfrac{\sqrt{2}}{\sqrt{2}}\\\\\\\\=~\dfrac{21\sqrt{(3-\sqrt{7}\,)\,\cdot\,2}}{\sqrt{2}^{\,2}}\\\\\\\\=~\boxed{\dfrac{21\sqrt{6-2\sqrt{7}}}{2}}$