Question

Simplify This Attachment​

Answer 1

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

Rationalize the denominator of $\frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}$ by multiplying numerator and denominator by $\tt \: \sqrt{2}-\sqrt{3}$

$\tt \: \frac{\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}+\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

Consider $\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right).$ Multiplication can be transformed into difference of squares using the rule: $\tt \: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$

$\tt \: \frac{\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}+\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \: \frac{\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)}{2-3}+\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \:</p><p>\frac{\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)}{2-3}+\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \:-\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

Rationalize the denominator of $\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}$ by multiplying numerator and denominator by $\tt \: \sqrt{6}-\sqrt{3}$

$\tt \: -\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{\left(\sqrt{6}+\sqrt{3}\right)\left(\sqrt{6}-\sqrt{3}\right)}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \: Consider \: \left(\sqrt{6}+\sqrt{3}\right)\left(\sqrt{6}-\sqrt{3}\right).$

Multiplication can be transformed into difference of squares using the rule: $\tt \: -\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{\left(\sqrt{6}+\sqrt{3}\right)\left(\sqrt{6}-\sqrt{3}\right)}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \\ \tt \: Consider \: \left(\sqrt{6}+\sqrt{3}\right)\left(\sqrt{6}-\sqrt{3}\right).$

$\tt \: -\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{6-3}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \: </p><p>-\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \: -\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}} \\ \\ \tt \: -\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)} \\ \\ \tt \:-\left(\sqrt{6}\sqrt{2}-\sqrt{6}\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \:-\left(\sqrt{2}\sqrt{3}\sqrt{2}-\sqrt{6}\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \: \tt \: -\left(2\sqrt{3}-\sqrt{6}\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \: -\left(2\sqrt{3}-\sqrt{3}\sqrt{2}\sqrt{3}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \ \tt \: -2\sqrt{3}-\left(-3\sqrt{2}\right)+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} </p><p> \\ \\ \tt \: -2\sqrt{3}+3\sqrt{2}+\frac{3\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \: -2\sqrt{3}+3\sqrt{2}+\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt\:-2\sqrt{3}+3\sqrt{2}+\sqrt{2}\sqrt{6}-\sqrt{2}\sqrt{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \: -2\sqrt{3}+3\sqrt{2}+\sqrt{2}\sqrt{2}\sqrt{3}-\sqrt{2}\sqrt{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \: </p><p>-2\sqrt{3}+3\sqrt{2}+2\sqrt{3}-\sqrt{2}\sqrt{3}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4} \\ \\ \tt \: 3\sqrt{2}-\sqrt{6}-\frac{4\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right)}{4}\\ \\ \tt \: 3\sqrt{2}-\sqrt{6}-\sqrt{3}\left(\sqrt{6}-\sqrt{2}\right) \\ \\ \tt \: </p><p>3\sqrt{2}-\sqrt{6}-\left(\sqrt{3}\sqrt{3}\sqrt{2}-\sqrt{3}\sqrt{2}\right) \\ \\ \tt \: 3\sqrt{2}-\sqrt{6}-\left(3\sqrt{2}-\sqrt{3}\sqrt{2}\right) \\ \\ \tt \: </p><p>3\sqrt{2}-\sqrt{6}-3\sqrt{2}-\left(-\sqrt{6}\right) \\ \\ \tt \: -\sqrt{6}+\sqrt{6}\\ \\ \tt \: 0$

$\sf \: @WildCat7083$