Question

Simplificando a expressão $<var>\frac{\sqrt{3}}{\sqrt{3}-1} - \frac{1}{\sqrt{3}} + \frac{2}{1+\sqrt{3}}</var>$ Vamos obter?

Answer 1

Primeiramente, observe que

 

$\text{mmc}(\sqrt{3}-1, \sqrt{3}, 1+\sqrt{3})=\sqrt{3}\cdot(\sqrt{3}-1)\cdot(\sqrt{3}+1)$

 

Desta maneira, temos:

 

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

 

$=\dfrac{3+3\sqrt{3}+1+6+2\sqrt{3}}{\sqrt{3}\cdot(\sqrt{3}-1)\cdot(\sqrt{3}+1)}$

 

$=\dfrac{10+5\sqrt{3}}{\sqrt{3}}=\dfrac{10\sqrt{3}}{3}+5$