\( \frac{15+\sqrt{18}}{3}= \) \( \frac{4-\sqrt{46}}{2}= \) nplex Fractions (Numerators) \( \begin{array}{l}\frac{65-12 \sqrt{75}}{5}= \\ \frac{5+\sqrt{18}}{\sqrt{3}}=\end{array} \)

Let’s simplify those fractions one by one! For \( \frac{15+\sqrt{18}}{3} \), you can rewrite \(\sqrt{18}\) as \(3\sqrt{2}\), giving you \( \frac{15 + 3\sqrt{2}}{3} = 5 + \sqrt{2} \). For \( \frac{4-\sqrt{46}}{2} \), divide both terms in the numerator by 2, yielding \( 2 - \frac{\sqrt{46}}{2} \). There's no simplification from here unless approximating \(\sqrt{46}\). Now, addressing the complex fractions in the numerators: For \( \frac{65 - 12\sqrt{75}}{5} \), note that \(\sqrt{75}\) is \(5\sqrt{3}\), so it becomes \( \frac{65 - 60\sqrt{3}}{5} = 13 - 12\sqrt{3} \). In the second one, \( \frac{5 + \sqrt{18}}{\sqrt{3}} \) gives \( \frac{5}{\sqrt{3}} + \frac{\sqrt{18}}{\sqrt{3}} = \frac{5}{\sqrt{3}} + \sqrt{6} \). To rationalize, multiply the first term by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get \( \frac{5\sqrt{3}}{3} + \sqrt{6} \). There you go! Simplifications can be a fun challenge!