To rationalize the denominator and simplify the given expression:
[tex]\[
\frac{4}{3 \sqrt{2} - 3}
\][/tex]
we will first multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 \sqrt{2} - 3\)[/tex] is [tex]\(3 \sqrt{2} + 3\)[/tex].
Let's proceed with this step-by-step.
1. Construct the conjugate of the denominator:
[tex]\[
3 \sqrt{2} + 3
\][/tex]
2. Multiply both the numerator and the denominator by the conjugate:
[tex]\[
\frac{4}{3 \sqrt{2} - 3} \times \frac{3 \sqrt{2} + 3}{3 \sqrt{2} + 3} = \frac{4 \cdot (3 \sqrt{2} + 3)}{(3 \sqrt{2} - 3) \cdot (3 \sqrt{2} + 3)}
\][/tex]
3. Simplify the numerator:
[tex]\[
4 \cdot (3 \sqrt{2} + 3) = 4 \cdot 3 \sqrt{2} + 4 \cdot 3 = 12 \sqrt{2} + 12
\][/tex]
4. Simplify the denominator using the difference of squares:
[tex]\[
(3 \sqrt{2} - 3) \cdot (3 \sqrt{2} + 3) = (3 \sqrt{2})^2 - 3^2 = 9 \cdot 2 - 9 = 18 - 9 = 9
\][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[
\frac{12 \sqrt{2} + 12}{9}
\][/tex]
6. Further simplify the expression if possible:
[tex]\[
\frac{12 (\sqrt{2} + 1)}{9} = \frac{4 (\sqrt{2} + 1)}{3}
\][/tex]
After performing the calculations, the final simplified result is:
[tex]\[
\frac{32.78460969082653}{18.0}
\][/tex]
This represents the rationalized and simplified form of the original expression:
[tex]\[
\frac{4}{3 \sqrt{2} - 3}
\][/tex]
