To express [tex]\(\frac{5-2 \sqrt{10}}{3 \sqrt{5}+\sqrt{2}}\)[/tex] in the form [tex]\(m \sqrt{2}+n \sqrt{5}\)[/tex], we follow these steps:
1. Identify the Conjugate:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 \sqrt{5} + \sqrt{2}\)[/tex] is [tex]\(3 \sqrt{5} - \sqrt{2}\)[/tex].
2. Multiply by the Conjugate:
Multiply the numerator and denominator by [tex]\(3 \sqrt{5} - \sqrt{2}\)[/tex]:
[tex]\[
\frac{5-2 \sqrt{10}}{3 \sqrt{5}+\sqrt{2}} \cdot \frac{3 \sqrt{5} - \sqrt{2}}{3 \sqrt{5} - \sqrt{2}}
\][/tex]
3. Expand the Numerator:
Expand the product in the numerator:
[tex]\[
(5 - 2 \sqrt{10})(3 \sqrt{5} - \sqrt{2})
\][/tex]
Distributing each term, we get:
[tex]\[
5 \cdot 3 \sqrt{5} - 5 \cdot \sqrt{2} - 2 \sqrt{10} \cdot 3 \sqrt{5} + 2 \sqrt{10} \cdot \sqrt{2}
\][/tex]
Simplifying each term, we have:
[tex]\[
15 \sqrt{5} - 5 \sqrt{2} - 6 \sqrt{50} + 2 \sqrt{20}
\][/tex]
Since [tex]\(\sqrt{50} = 5 \sqrt{2}\)[/tex] and [tex]\(\sqrt{20} = 2 \sqrt{5}\)[/tex], we can further simplify:
[tex]\[
15 \sqrt{5} - 5 \sqrt{2} - 6 \cdot 5 \sqrt{2} + 2 \cdot 2 \sqrt{5}
\][/tex]
[tex]\[
15 \sqrt{5} - 5 \sqrt{2} - 30 \sqrt{2} + 4 \sqrt{5}
\][/tex]
Collecting like terms, we get:
[tex]\[
19 \sqrt{5} - 35 \sqrt{2}
\][/tex]
4. Expand the Denominator:
Expand the product in the denominator:
[tex]\[
(3 \sqrt{5} + \sqrt{2})(3 \sqrt{5} - \sqrt{2})
\][/tex]
This is a difference of squares, so it simplifies to:
[tex]\[
(3 \sqrt{5})^2 - (\sqrt{2})^2
\][/tex]
Calculating each term, we get:
[tex]\[
9 \cdot 5 - 2
\][/tex]
[tex]\[
45 - 2 = 43
\][/tex]
5. Form the Rationalized Expression:
We now have:
[tex]\[
\frac{19 \sqrt{5} - 35 \sqrt{2}}{43}
\][/tex]
6. Express in the Desired Form:
Separate the terms to express it in the form [tex]\(m \sqrt{2} + n \sqrt{5}\)[/tex]:
[tex]\[
\frac{19 \sqrt{5} - 35 \sqrt{2}}{43} = \frac{19 \sqrt{5}}{43} - \frac{35 \sqrt{2}}{43}
\][/tex]
Thus, the expression [tex]\(\frac{5-2 \sqrt{10}}{3 \sqrt{5}+\sqrt{2}}\)[/tex] in the form [tex]\(m \sqrt{2}+n \sqrt{5}\)[/tex] is:
[tex]\[
m = -\frac{35}{43}, \quad n = \frac{19}{43}
\][/tex]
So, we have:
[tex]\[
\boxed{\frac{5-2 \sqrt{10}}{3 \sqrt{5}+\sqrt{2}} = -\frac{35 \sqrt{2}}{43} + \frac{19 \sqrt{5}}{43}}
\][/tex]
