Answer :
Step-by-step explanation:
To rationalize the denominator of the expression \(\frac{2\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} + 3\sqrt{2}}\), we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2\sqrt{3} + 3\sqrt{2}\) is \(2\sqrt{3} - 3\sqrt{2}\).
Here are the steps to rationalize the denominator:
1. **Multiply the numerator and the denominator by the conjugate of the denominator**:
\[
\frac{2\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} + 3\sqrt{2}} \times \frac{2\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 3\sqrt{2}}
\]
2. **Simplify the numerator and the denominator separately**:
**Numerator**:
\[
(2\sqrt{3} - 3\sqrt{2})(2\sqrt{3} - 3\sqrt{2})
\]
\[
= (2\sqrt{3})^2 - 2 \cdot 2\sqrt{3} \cdot 3\sqrt{2} + (3\sqrt{2})^2
\]
\[
= 4 \cdot 3 - 2 \cdot 2 \cdot 3 \cdot \sqrt{6} + 9 \cdot 2
\]
\[
= 12 - 12\sqrt{6} + 18
\]
\[
= 30 - 12\sqrt{6}
\]
**Denominator**:
\[
(2\sqrt{3} + 3\sqrt{2})(2\sqrt{3} - 3\sqrt{2})
\]
\[
= (2\sqrt{3})^2 - (3\sqrt{2})^2
\]
\[
= 4 \cdot 3 - 9 \cdot 2
\]
\[
= 12 - 18
\]
\[
= -6
\]
3. **Combine the results**:
\[
\frac{30 - 12\sqrt{6}}{-6}
\]
4. **Simplify the fraction** by dividing both terms in the numerator by \(-6\):
\[
= \frac{30}{-6} - \frac{12\sqrt{6}}{-6}
\]
\[
= -5 + 2\sqrt{6}
\]
Therefore, the rationalized form of the expression is:
\[
-5 + 2\sqrt{6}
\]