1. Express \( \frac{3 \sqrt{2}-2 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}} \) in the form \( p+q \sqrt{r} \)

To express \( \frac{3 \sqrt{2}-2 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}} \) in the form \( p + q \sqrt{r} \), we'll multiply the numerator and the denominator by the conjugate of the denominator, which is \( 3 \sqrt{2} - 2 \sqrt{3} \). Let's perform the multiplication: \[ \frac{(3\sqrt{2}-2\sqrt{3})(3\sqrt{2}-2\sqrt{3})}{(3\sqrt{2}+2\sqrt{3})(3\sqrt{2}-2\sqrt{3})} \] First, let’s compute the denominator: \[ (3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6 \] Now for the numerator: \[ (3\sqrt{2}-2\sqrt{3})^2 = (3\sqrt{2})^2 - 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 \] \[ = 18 - 12\sqrt{6} + 12 = 30 - 12\sqrt{6} \] Putting it all together, we have: \[ \frac{30 - 12\sqrt{6}}{6} = 5 - 2\sqrt{6} \] Thus, in the form \( p + q\sqrt{r} \), we have \( p = 5\), \( q = -2\), and \( r = 6\). The final answer is: \[ \boxed{5 - 2\sqrt{6}} \]