nator and Simplify: (2sqrt (5)-sqrt (6))/(3-sqrt (5))

To rationalize and simplify the expression $\frac {2\sqrt {5}-\sqrt {6}}{3-\sqrt {5}}$, we can multiply both the numerator and denominator by the conjugate of the denominator, which is $3+\sqrt{5}$.<br /><br />So, we have:<br /><br />$\frac {2\sqrt {5}-\sqrt {6}}{3-\sqrt {5}} \times \frac {3+\sqrt {5}}{3+\sqrt {5}}$<br /><br />Multiplying the numerators together, we get:<br /><br />$(2\sqrt {5}-\sqrt {6})(3+\sqrt {5})$<br /><br />Expanding this, we have:<br /><br />$6\sqrt {5} + 2\sqrt {25} - 3\sqrt {6} - \sqrt {30}$<br /><br />Simplifying further, we get:<br /><br />$6\sqrt {5} + 10 - 3\sqrt {6} - \sqrt {30}$<br /><br />Now, let's multiply the denominators together:<br /><br />$(3-\sqrt {5})(3+\sqrt {5})$<br /><br />Using the difference of squares formula, we get:<br /><br />$9 - 5$<br /><br />Simplifying this, we have:<br /><br />$4$<br /><br />So, the rationalized and simplified expression is:<br /><br />$\frac {6\sqrt {5} + 10 - 3\sqrt {6} - \sqrt {30}}{4}$<br /><br />We can further simplify this by dividing each term in the numerator by 4:<br /><br />$\frac {3\sqrt {5} + 5 - \frac {3\sqrt {6}}{2} - \frac {\sqrt {30}}{4}}{1}$<br /><br />Therefore, the final simplified expression is:<br /><br />$\frac {3\sqrt {5} + 5 - \frac {3\sqrt {6}}{2} - \frac {\sqrt {30}}{4}}{1}$