Steps to Rationalize the Denominator and Simplify
1. Rationalize the Denominator:
To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of the denominator is the same expression but with the opposite sign between the terms.
2. First Fraction:
$\frac{3\sqrt{2}}{\sqrt{3}+\sqrt{6}}$
Multiply numerator and denominator by the conjugate of the denominator:
$\frac{3\sqrt{2}(\sqrt{3}-\sqrt{6})}{(\sqrt{3}+\sqrt{6})(\sqrt{3}-\sqrt{6})}$
Simplify the denominator:
$3\sqrt{6}-3\sqrt{3}$
3. Second Fraction:
$\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}}$
Multiply numerator and denominator by the conjugate of the denominator:
$\frac{4\sqrt{3}(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})}$
Simplify the denominator:
$4\sqrt{6}-4\sqrt{2}$
4. Third Fraction:
$\frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}$
Multiply numerator and denominator by the conjugate of the denominator:
$\frac{\sqrt{6}(\sqrt{2}-\sqrt{3})}{(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})}$
Simplify the denominator:
$\sqrt{2}-\sqrt{3}$
5. Final Answer:
Putting all the fractions back together, we get:
$\frac{3\sqrt{2}(\sqrt{3}-\sqrt{6})}{3\sqrt{6}-3\sqrt{3}} - \frac{4\sqrt{3}(\sqrt{6}-\sqrt{2})}{4\sqrt{6}-4\sqrt{2}} + \frac{\sqrt{6}(\sqrt{2}-\sqrt{3})}{\sqrt{2}-\sqrt{3}}$
This can be further simplified by combining like terms in the numerator and denominator.