Simplifying Radical Expressions x and y

First, simplify the expression for \(x\) by rationalizing the denominator. Multiply the numerator and denominator by the conjugate of the denominator, which is \((\sqrt{6} + \sqrt{5})\):

\(x = \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}} \times \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} + \sqrt{5}} = \frac{(\sqrt{6} + \sqrt{5})^2}{(\sqrt{6})^2 - (\sqrt{5})^2}\) $ \(x = \frac{(\sqrt{6})^2 + 2(\sqrt{6})(\sqrt{5}) + (\sqrt{5})^2}{6 - 5} = \frac{6 + 2\sqrt{30} + 5}{1} = 11 + 2\sqrt{30}\)

Next, simplify the expression for \(y\) by rationalizing the denominator. Multiply the numerator and denominator by the conjugate of the denominator, which is \((\sqrt{6} - \sqrt{5})\):

\(y = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} = \frac{(\sqrt{6} - \sqrt{5})^2}{(\sqrt{6})^2 - (\sqrt{5})^2}\) $ \(y = \frac{(\sqrt{6})^2 - 2(\sqrt{6})(\sqrt{5}) + (\sqrt{5})^2}{6 - 5} = \frac{6 - 2\sqrt{30} + 5}{1} = 11 - 2\sqrt{30}\)

Calculating Sum and Difference

To efficiently calculate \((x^2 - y^2)\), we use the difference of squares identity: \(x^2 - y^2 = (x + y)(x - y)\). Calculate the sum \((x + y)\):

\(x + y = (11 + 2\sqrt{30}) + (11 - 2\sqrt{30}) = 11 + 11 = 22\) $

Calculate the difference \((x - y)\):

\(x - y = (11 + 2\sqrt{30}) - (11 - 2\sqrt{30}) = 11 + 2\sqrt{30} - 11 + 2\sqrt{30} = 4\sqrt{30}\) $

Final Calculation of x^2 - y^2

Substitute the calculated values of \((x + y)\) and \((x - y)\) into the difference of squares formula:

\(x^2 - y^2 = (x + y)(x - y) = (22)(4\sqrt{30})\) $

\(x^2 - y^2 = 88\sqrt{30}\) $