To solve the problem [tex]\( x - y \sqrt{6} = \frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}} \)[/tex] and find the value of [tex]\( x - y \)[/tex], we can follow these detailed steps:
1. Rationalizing the Denominator:
We start by rationalizing the denominator of the given expression:
[tex]\[
\frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}}
\][/tex]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{(5 - 2\sqrt{6})(5 - 2\sqrt{6})}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})}
\][/tex]
2. Simplifying the Denominator:
The denominator simplifies as follows:
[tex]\[
(5 + 2\sqrt{6})(5 - 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1
\][/tex]
So the denominator becomes [tex]\(1\)[/tex].
3. Simplifying the Numerator:
The numerator is:
[tex]\[
(5 - 2\sqrt{6})^2 = 5^2 - 2 \cdot 5 \cdot 2\sqrt{6} + (2\sqrt{6})^2
\][/tex]
Simplifying further,
[tex]\[
25 - 20\sqrt{6} + 24 = 49 - 20\sqrt{6}
\][/tex]
4. Combining Results:
So the given expression simplifies to:
[tex]\[
x - y\sqrt{6} = 49 - 20\sqrt{6}
\][/tex]
By comparing this to [tex]\( x - y\sqrt{6} \)[/tex], we can see that:
[tex]\[
x = 49
\][/tex]
and
[tex]\[
y = -20
\][/tex]
5. Finding [tex]\( x - y \)[/tex]:
Finally, we calculate:
[tex]\[
x - y = 49 - (-20) = 49 + 20 = 69
\][/tex]
Thus, the value of [tex]\( x - y \)[/tex] is:
[tex]\[
\boxed{29}
\][/tex]
