Right, let's go. To rationalize the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $5\sqrt{2} - 2\sqrt{3}$ is $5\sqrt{2} + 2\sqrt{3}$. Step 1: Multiply by the conjugate. $$ \frac{2\sqrt{2} + 3\sqrt{3}}{5\sqrt{2} - 2\sqrt{3}} \times \frac{5\sqrt{2} + 2\sqrt{3}}{5\sqrt{2} + 2\sqrt{3}} $$ Step 2: Expand the numerator and the denominator. For the numerator: $(2\sqrt{2} + 3\sqrt{3})(5\sqrt{2} + 2\sqrt{3})$ $$ (2\sqrt{2})(5\sqrt{2}) + (2\sqrt{2})(2\sqrt{3}) + (3\sqrt{3})(5\sqrt{2}) + (3\sqrt{3})(2\sqrt{3}) $$ $$ 10(\sqrt{2}\sqrt{2}) + 4(\sqrt{2}\sqrt{3}) + 15(\sqrt{3}\sqrt{2}) + 6(\sqrt{3}\sqrt{3}) $$ $$ 10(2) + 4\sqrt{6} + 15\sqrt{6} + 6(3) $$ $$ 20 + 19\sqrt{6} + 18 $$ $$ 38 + 19\sqrt{6} $$ For the denominator: $(5\sqrt{2} - 2\sqrt{3})(5\sqrt{2} + 2\sqrt{3})$ This is in the form $(a-b)(a+b) = a^2 - b^2$. $$ (5\sqrt{2})^2 - (2\sqrt{3})^2 $$ $$ (5^2 \times (\sqrt{2})^2) - (2^2 \times (\sqrt{3})^2) $$ $$ (25 \times 2) - (4 \times 3) $$ $$ 50 - 12 $$ $$ 38 $$ Step 3: Combine the simplified numerator and denominator. $$ \frac{38 + 19\sqrt{6}}{38} $$ Step 4: Simplify the fraction by dividing each term in the numerator by the denominator. $$ \frac{38}{38} + \frac{19\sqrt{6}}{38} $$ $$ 1 + \frac{\sqrt{6}}{2} $$ The rationalized expression is: $$ \boxed{1 + \frac{\sqrt{6}}{2}} $$ Send me the next one 📸