1. The problem shows three pairs of numbers connected by arrows with numbers in parentheses, indicating a transformation or operation between the numbers. 2. Let's analyze the first pair: $\frac{4}{9}$ connected leftward by arrow labeled $(2)$ to $\frac{2}{3}$. 3. The arrow pointing left suggests the operation is applied to $\frac{2}{3}$ to get $\frac{4}{9}$. 4. To find the operation, check if multiplying or dividing $\frac{2}{3}$ by $2$ gives $\frac{4}{9}$. 5. Multiply $\frac{2}{3}$ by $2$: $$\frac{2}{3} \times 2 = \frac{4}{3}$$ which is not $\frac{4}{9}$. 6. Divide $\frac{2}{3}$ by $2$: $$\frac{2}{3} \div 2 = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}$$ which is not $\frac{4}{9}$. 7. Try squaring $\frac{2}{3}$: $$\left(\frac{2}{3}\right)^2 = \frac{4}{9}$$ which matches the left number. 8. So the arrow labeled $(2)$ means "raise to the power of 2". 9. Check the second pair: $36 \frac{1}{4}$ connected leftward by arrow $(2)$ to $6 \frac{1}{2}$. 10. Convert mixed numbers to improper fractions: $$36 \frac{1}{4} = \frac{145}{4}$$ $$6 \frac{1}{2} = \frac{13}{2}$$ 11. Square $\frac{13}{2}$: $$\left(\frac{13}{2}\right)^2 = \frac{169}{4}$$ which is close but not equal to $\frac{145}{4}$. 12. Check if squaring is approximate or if the arrow means something else. 13. Check if multiplying $6 \frac{1}{2}$ by $2$ gives $36 \frac{1}{4}$: $$\frac{13}{2} \times 2 = 13$$ not $\frac{145}{4}$. 14. Check if squaring minus something equals $\frac{145}{4}$: $$\frac{169}{4} - \frac{24}{4} = \frac{145}{4}$$ 15. Possibly the arrow means "square and subtract 6" (since $6 = \frac{24}{4}$), but this is speculative. 16. Check the third pair: $8 \frac{1}{27}$ connected leftward by arrow $(3)$ to $2 \frac{2}{3}$. 17. Convert to improper fractions: $$8 \frac{1}{27} = \frac{217}{27}$$ $$2 \frac{2}{3} = \frac{8}{3}$$ 18. Cube $\frac{8}{3}$: $$\left(\frac{8}{3}\right)^3 = \frac{512}{27}$$ which is not $\frac{217}{27}$. 19. Cube root $\frac{217}{27}$: $$\sqrt[3]{\frac{217}{27}} = \frac{\sqrt[3]{217}}{3}$$ which is approximately $2.99$ not $\frac{8}{3} = 2.67$. 20. The first pair clearly shows the arrow labeled $(2)$ means "square". 21. Therefore, the operation indicated by the arrow labeled $(2)$ is raising to the power of 2. Final answer: The arrow labeled $(2)$ means "raise to the power of 2" (square).