Question

simplify --------------

2root6 divided by root2 +root 3 + 6 root2 divided by root 6 + root 3 - 8 root 3 divided by root6 +root 2

Answer 1

Answer:

Given: $\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

Consider,

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

Rationalize the denominator of each term,

$=\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}\times\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}\times\frac{\sqrt{6}-\sqrt{3}}{\sqrt{6}-\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}\times\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}-\sqrt{2}}$

Use identity in Denominator a² - b² = ( a + b ) ( a - b )

$=\frac{4\sqrt{3}-6\sqrt{2}}{2-3}+\frac{12\sqrt{3}-6\sqrt{6}}{6-3}-\frac{24\sqrt{2}-8\sqrt{6}}{6-2}$

$=-4\sqrt{3}+6\sqrt{2}+4\sqrt{3}-2\sqrt{6}-6\sqrt{2}+2\sqrt{6}$

$=0$

Therefore, $\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}=0$

Answer 2

Simplifying $\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

According to the question we get,

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

Rationalizing  the denominator we get,

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

=$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}\times\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}\times\frac{\sqrt{6}-\sqrt{3}}{\sqrt{6}-\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}\times\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}-\sqrt{2}}$

We know a² - b² = ( a + b ) ( a - b )

Thus using this we get,

$\frac{4\sqrt{3}-6\sqrt{2}}{2-3}+\frac{12\sqrt{3}-6\sqrt{6}}{6-3}-\frac{24\sqrt{2}-8\sqrt{6}}{6-2}$

  Now simplifying the equation we get,

$-4\sqrt{3}+6\sqrt{2}+4\sqrt{3}-2\sqrt{6}-6\sqrt{2}+2\sqrt{6}$

All the terms are thus cross out.

Thus we get,

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

=0