\[\text{Let } x = \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}}, \text{ and } y = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} + \sqrt{5}}\]
\(88\sqrt{30}\)
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:
\(x^2 - y^2 = (x + y)(x - y) = (22)(4\sqrt{30})\) $
\(x^2 - y^2 = 88\sqrt{30}\) $
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