Question

Simplify:$\frac{6}{2\sqrt{3}-\sqrt{6}}+\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}-\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$.

Answer 1

Step 1: Rationalize $\frac{6}{2\sqrt{3}-\sqrt{6}}$.

Multiply numerator and denominator of $\frac{6}{2\sqrt{3}-\sqrt{6}}$by $2\sqrt{3}+\sqrt{6}$.

Therefore,

$\begin{array}{c}\frac{6}{2\sqrt{3}-\sqrt{6}}\times \frac{2\sqrt{3}+\sqrt{6}}{2\sqrt{3}+\sqrt{6}}=\frac{6(2\sqrt{3}+\sqrt{6})}{{\left(2\sqrt{3}\right)}^2-{\left(\sqrt{6}\right)}^2}\\ =\frac{6(2\sqrt{3}+\sqrt{6})}{12-6}\\ =\frac{6(2\sqrt{3}+\sqrt{6})}{6}\\ =2\sqrt{3}+\sqrt{6}\end{array}$

Step 2: Rationalize $\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$.

Multiply numerator and denominator of $\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$by $\sqrt{3}-\sqrt{2}$.

Therefore,

$\begin{array}{c}\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}\times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\frac{\sqrt{6}(\sqrt{3}-\sqrt{2})}{{\left(\sqrt{3}\right)}^2-{\left(\sqrt{2}\right)}^2}\\ =\frac{\sqrt{6}(\sqrt{3}-\sqrt{2})}{3-2}\\ =\frac{\sqrt{6}(\sqrt{3}-\sqrt{2})}{1}\\ =\sqrt{6}(\sqrt{3}-\sqrt{2})\\ =3\sqrt{2}-2\sqrt{3}\end{array}$

Step 3: Rationalize $\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$.

Multiply numerator and denominator of $\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$by $\sqrt{6}+\sqrt{2}$.

Therefore,

$\begin{array}{c}\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}\times \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}}=\frac{4\sqrt{3}(\sqrt{6}-\sqrt{2})}{{\left(\sqrt{6}\right)}^2-{\left(\sqrt{2}\right)}^2}\\ =\frac{4\sqrt{3}(\sqrt{6}-\sqrt{2})}{6-2}\\ =\frac{4\sqrt{3}(\sqrt{6}-\sqrt{2})}{4}\\ =\sqrt{3}(\sqrt{6}-\sqrt{2})\\ =3\sqrt{2}-\sqrt{6}\end{array}$

Step 4: Perform mathematical operations.

Perform the required mathematical operation and deduce the answer.

Therefore,

$\begin{array}{c}\frac{6}{2\sqrt{3}-\sqrt{6}}+\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}-\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}=(2\sqrt{3}+\sqrt{6})+(3\sqrt{2}-2\sqrt{3})-(3\sqrt{2}-\sqrt{6})\\ =2\sqrt{3}+\sqrt{6}+3\sqrt{2}-2\sqrt{3}-3\sqrt{2}-\sqrt{6}\\ =2\sqrt{3}-2\sqrt{3}+\sqrt{6}-\sqrt{6}+3\sqrt{2}-3\sqrt{2}\\ =0\end{array}$

Hence, the value of $\frac{6}{2\sqrt{3}-\sqrt{6}}+\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}-\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$is $0$.