Sure, let's solve the fraction [tex]\(\frac{4}{3-\sqrt{2}}\)[/tex].
### Step-by-Step Solution:
1. Original Fraction:
[tex]\[
\frac{4}{3 - \sqrt{2}}
\][/tex]
2. Rationalizing the Denominator:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\( 3 - \sqrt{2} \)[/tex] is [tex]\( 3 + \sqrt{2} \)[/tex].
Thus, we multiply by [tex]\( \frac{3 + \sqrt{2}}{3 + \sqrt{2}} \)[/tex]:
[tex]\[
\frac{4}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}}
\][/tex]
3. Simplifying the Numerator:
[tex]\[
4 (3 + \sqrt{2}) = 4 \cdot 3 + 4 \cdot \sqrt{2} = 12 + 4\sqrt{2}
\][/tex]
4. Simplifying the Denominator:
[tex]\[
(3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7
\][/tex]
5. Combined Fraction:
[tex]\[
\frac{12 + 4\sqrt{2}}{7}
\][/tex]
6. Simplifying the Expression:
Breaking it down further:
[tex]\[
\frac{12 + 4\sqrt{2}}{7} = \frac{12}{7} + \frac{4\sqrt{2}}{7}
\][/tex]
### Numerical Values:
- Numerator: [tex]\( 12 + 4\sqrt{2} \)[/tex]
- Denominator: [tex]\( 7 \)[/tex]
- The simplified fractional result in a numerical form is approximately [tex]\( 2.52240774992748 \)[/tex]
Therefore, the fraction [tex]\(\frac{4}{3 - \sqrt{2}}\)[/tex] simplifies to approximately [tex]\(\frac{12 + 4\sqrt{2}}{7}\)[/tex] or numerically [tex]\(\approx 2.52240774992748\)[/tex].
