Question

Fuvest(adaptado)
Racionalize o denominador
1/ 1+√2-√3

Answer 1

$\frac{1}{1+\sqrt{2}-\sqrt{3}}=\frac{1}{(1+\sqrt{2})-\sqrt{3}}$

Multiplicando pelo conjugado do denominador temos

$\frac{1}{1+\sqrt{2}-\sqrt{3}}=\frac{1}{((1+\sqrt{2})-\sqrt{3})}.\frac{((1+\sqrt{2})+\sqrt{3})}{(1+\sqrt{2})+\sqrt{3})}$

$\frac{(1+\sqrt{2})+\sqrt{3}}{{(1+\sqrt{2})}^{2}-{(\sqrt{3})}^{2}}$

$\frac{(1+\sqrt{2})+\sqrt{3}}{1+2\sqrt{2}+2-3}$

$\frac{(1+\sqrt{2})+\sqrt{3}}{\cancel{1}+2\sqrt{2}-\cancel{1}}$

$\frac{(1+\sqrt{2})+\sqrt{3}}{2\sqrt{2}}$

$\frac{(1+\sqrt{2}+\sqrt{3})}{2\sqrt{2}}.\frac{\sqrt{2}}{\sqrt{2}}$

$\frac{(\sqrt{2}+2+\sqrt{6})}{4}$

$\boxed{\boxed{\frac{1}{1+\sqrt{2}-\sqrt{3}}=\frac{(\sqrt{2}+2+\sqrt{6})}{4}}}$