1. **State the problem:** We are given a hierarchical tree of mathematical expressions and asked to evaluate the root node labeled "5" based on the expressions and their relationships. 2. **Understand the structure:** The tree has multiple levels with expressions connected by division and multiplication. We need to evaluate from the bottom up. 3. **Evaluate the bottom leaves:** - Leaf 1: $$\frac{x^4 + 5x^3}{3}$$ - Leaf 2: $$\frac{1}{x^2 - y^2}$$ - Leaf 3: $$\frac{1}{(x - y)^2}$$ - Leaf 4: $$\frac{9x^2}{8}$$ 4. **Evaluate the next level up:** - Left child of "24x^3": $$\frac{2x}{2x + 5}$$ - Right child of "24x^3": $$\frac{1}{2}(x + 1)^2$$ - Child of "(2x^2 + 2)/(x + 1)": $$\frac{8x}{3(x + 2)^5}$$ 5. **Evaluate the middle nodes:** - Left node under root: "24x^3" - Right node under root: $$\frac{2x^2 + 2}{x + 1}$$ 6. **Combine the expressions:** - The root node "5" is the top node, presumably representing the value or result of the entire tree. 7. **Summary:** Without specific values for variables $x$ and $y$, the expressions cannot be numerically evaluated. The tree shows how complex expressions are built from simpler ones. **Final answer:** The root node is labeled "5" as given, representing the value or label of the entire tree structure. If you want to evaluate for specific $x$ and $y$ values, please provide them.