Rationalize\:the\:denominator\:and\:\:simplify\:\frac{7-\sqrt{6}}{5+3\sqrt{6}}
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of 5 + 3√6 is 5 - 3√6.
So, multiplying the numerator and denominator by 5 - 3√6, we have:
\[\frac{7 - \sqrt{6}}{5 + 3\sqrt{6}} \times \frac{5 - 3\sqrt{6}}{5 - 3\sqrt{6}} = \frac{(7 - \sqrt{6})(5 - 3\sqrt{6})}{(5 + 3\sqrt{6})(5 - 3\sqrt{6})}\]
Expanding the numerator and denominator, we get:
\[\frac{35 - 3\sqrt{6} - 21\sqrt{6} + 18}{25 - 9(6)}\]
Simplifying further, we have:
\[\frac{35 - 21\sqrt{6} - 3\sqrt{6} + 18}{25 - 54}\]
This becomes:
\[\frac{53 - 24\sqrt{6}}{-29} = \boxed{\frac{-53 + 24\sqrt{6}}{29}}\]