Let's solve the given problem step by step.
We are given:
[tex]\[ p = 3 - 2 \sqrt{2} \][/tex]
[tex]\[ q = 2 - \sqrt{2} \][/tex]
We need to find the value of [tex]\(\frac{p+q}{p-q}\)[/tex].
First, compute [tex]\( p + q \)[/tex]:
[tex]\[
p + q = (3 - 2\sqrt{2}) + (2 - \sqrt{2}) = 3 + 2 - 2\sqrt{2} - \sqrt{2} = 5 - 3\sqrt{2}
\][/tex]
Next, compute [tex]\( p - q \)[/tex]:
[tex]\[
p - q = (3 - 2\sqrt{2}) - (2 - \sqrt{2}) = 3 - 2 - 2\sqrt{2} + \sqrt{2} = 1 - \sqrt{2}
\][/tex]
Now, we have:
[tex]\[
\frac{p+q}{p-q} = \frac{5 - 3\sqrt{2}}{1 - \sqrt{2}}
\][/tex]
To simplify this expression, we rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator:
The conjugate of [tex]\(1 - \sqrt{2}\)[/tex] is [tex]\(1 + \sqrt{2}\)[/tex]. So, we multiply both the numerator and the denominator by [tex]\(1 + \sqrt{2}\)[/tex]:
[tex]\[
\frac{(5 - 3\sqrt{2})(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})}
\][/tex]
First, simplify the denominator:
[tex]\[
(1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1
\][/tex]
Next, we simplify the numerator:
[tex]\[
(5 - 3\sqrt{2})(1 + \sqrt{2}) = 5 \cdot 1 + 5 \cdot \sqrt{2} - 3 \sqrt{2} \cdot 1 - 3\sqrt{2} \cdot \sqrt{2} = 5 + 5\sqrt{2} - 3\sqrt{2} - 3 \cdot 2
\][/tex]
[tex]\[
= 5 + 2\sqrt{2} - 6 = -1 + 2\sqrt{2}
\][/tex]
Therefore,
[tex]\[
\frac{(5 - 3\sqrt{2})(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})} = \frac{-1 + 2\sqrt{2}}{-1} = 1 - 2\sqrt{2}
\][/tex]
So, the value of [tex]\(\frac{p+q}{p-q}\)[/tex] in the form [tex]\(m + n\sqrt{2}\)[/tex] is:
[tex]\[
\boxed{1 - 2\sqrt{2}}
\][/tex]
Thus, we have [tex]\( m = 1 \)[/tex] and [tex]\( n = -2 \)[/tex].
