Answer :

GinnyAnswer
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]