To rationalize the expression [tex]\(\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{2}}\)[/tex], we follow these steps:
1. Identify the Conjugate:
For the denominator [tex]\(3 - 2\sqrt{2}\)[/tex], the conjugate is [tex]\(3 + 2\sqrt{2}\)[/tex].
2. Multiply the Numerator and Denominator by the Conjugate:
Multiply both the numerator and the denominator by the conjugate of the denominator to remove the irrational part from the denominator.
[tex]\[
\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}}
\][/tex]
3. Expand the Numerator:
[tex]\[
(3 + 2\sqrt{3})(3 + 2\sqrt{2})
\][/tex]
Using the distributive property:
[tex]\[
= 3 \cdot 3 + 3 \cdot 2\sqrt{2} + 2\sqrt{3} \cdot 3 + 2\sqrt{3} \cdot 2\sqrt{2}
\][/tex]
Simplify the terms:
[tex]\[
= 9 + 6\sqrt{2} + 6\sqrt{3} + 4\sqrt{6}
\][/tex]
4. Simplify the Denominator:
[tex]\[
(3 - 2\sqrt{2})(3 + 2\sqrt{2})
\][/tex]
This is a difference of squares:
[tex]\[
= 3^2 - (2\sqrt{2})^2
\][/tex]
Simplify the terms:
[tex]\[
= 9 - 4 \cdot 2
\][/tex]
[tex]\[
= 9 - 8 = 1
\][/tex]
5. Write the Rationalized Expression:
The numerator simplifies to approximately 37.67554519078455 and the denominator simplifies to approximately 1. Therefore, the rationalized expression is:
[tex]\[
\frac{3 + 2\sqrt{3})(3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} = 37.67554519078455
\][/tex]
The final answer is very close to:
[tex]\[
\boxed{37.67554519078461}
\][/tex]
