\[\text{Let } x = \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}}, \text{ and } y = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} + \sqrt{5}}\]
481
We are given the expressions for \(x\) and \(y\):
- \(x = \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}}\)
- \(y = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} + \sqrt{5}}\)
We need to find the value of \(x^2 - xy + y^2\).
Simplify Expressions for x and y
To simplify \(x\), we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{6} + \sqrt{5}\):
\(x = \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}} \times \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} + \sqrt{5}}\)
\(x = \frac{(\sqrt{6} + \sqrt{5})^2}{(\sqrt{6})^2 - (\sqrt{5})^2}\)
\(x = \frac{6 + 2(\sqrt{6})(\sqrt{5}) + 5}{6 - 5}\)
\(x = \frac{11 + 2\sqrt{30}}{1}\)
\(x = 11 + 2\sqrt{30}\)
Similarly, to simplify \(y\), we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{6} - \sqrt{5}\):
\(y = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}}\)
\(y = \frac{(\sqrt{6} - \sqrt{5})^2}{(\sqrt{6})^2 - (\sqrt{5})^2}\)
\(y = \frac{6 - 2(\sqrt{6})(\sqrt{5}) + 5}{6 - 5}\)
\(y = \frac{11 - 2\sqrt{30}}{1}\)
\(y = 11 - 2\sqrt{30}\)
Calculate Intermediate Values
We can express \(x^2 - xy + y^2\) using \(x+y\) and \(xy\). Note that \(x^2 - xy + y^2 = (x^2 + 2xy + y^2) - 3xy = (x+y)^2 - 3xy\).
First, calculate \(x+y\):
\(x + y = (11 + 2\sqrt{30}) + (11 - 2\sqrt{30})\)
\(x + y = 11 + 11 + 2\sqrt{30} - 2\sqrt{30}\)
\(x + y = 22\)
Next, calculate \(xy\):
\(xy = (11 + 2\sqrt{30})(11 - 2\sqrt{30})\)
This is in the form \((a+b)(a-b) = a^2 - b^2\).
\(xy = (11)^2 - (2\sqrt{30})^2\)
\(xy = 121 - (4 \times 30)\)
\(xy = 121 - 120\)
\(xy = 1\)
Calculate Final Expression Value
Now substitute the values of \(x+y\) and \(xy\) into the expression \((x+y)^2 - 3xy\):
\(x^2 - xy + y^2 = (x+y)^2 - 3xy\)
\(= (22)^2 - 3(1)\)
\(= 484 - 3\)
\(= 481\)
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