To solve the expression [tex]\(\frac{10 - \sqrt{18}}{\sqrt{2}}\)[/tex] and express it in the form [tex]\(a + b\sqrt{2}\)[/tex]:
1. Rewrite the expression:
[tex]\[
\frac{10 - \sqrt{18}}{\sqrt{2}}
\][/tex]
2. Rationalize the denominator by multiplying both the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\frac{(10 - \sqrt{18}) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}
\][/tex]
Since [tex]\(\sqrt{2} \cdot \sqrt{2} = 2\)[/tex], the expression simplifies to:
[tex]\[
\frac{(10 - \sqrt{18}) \cdot \sqrt{2}}{2}
\][/tex]
3. Distribute [tex]\(\sqrt{2}\)[/tex] in the numerator:
[tex]\[
\frac{10\sqrt{2} - \sqrt{18}\sqrt{2}}{2}
\][/tex]
Since [tex]\(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\)[/tex], substitute [tex]\(3\sqrt{2}\)[/tex] for [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\frac{10\sqrt{2} - 3\sqrt{2} \cdot \sqrt{2}}{2}
\][/tex]
4. Simplify the expression inside the numerator:
[tex]\[
\frac{10\sqrt{2} - 3 \cdot 2}{2}
\][/tex]
Since [tex]\(3 \cdot 2 = 6\)[/tex]:
[tex]\[
\frac{10\sqrt{2} - 6}{2}
\][/tex]
5. Separate the fraction into two parts:
[tex]\[
\frac{10\sqrt{2}}{2} - \frac{6}{2}
\][/tex]
6. Simplify each part:
[tex]\[
5\sqrt{2} - 3
\][/tex]
So, the simplified expression [tex]\(\frac{10 - \sqrt{18}}{\sqrt{2}}\)[/tex] in the form [tex]\(a + b\sqrt{2}\)[/tex] is:
[tex]\[
a = -3 \quad \text{and} \quad b = 5
\][/tex]
Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
\boxed{(a, b) = (-3, 5)}
\][/tex]
