1. **Stating the problem:** We want to express the given expression $$\frac{(-3 \times -5)}{(5 - 3 + (+3 \times -8))} + (6 - 2 \times -1 + 3)$$ in lexicographic, postfix, and prefix notations. 2. **Recall the expression:** $$\frac{(-3 \times -5)}{(5 - 3 + (+3 \times -8))} + (6 - 2 \times -1 + 3)$$ 3. **Lexicographic notation (infix):** This is the original expression with parentheses showing operation order: $$(((-3) \times (-5)) / (5 - 3 + ((+3) \times (-8)))) + (6 - (2 \times (-1)) + 3)$$ 4. **Postfix notation (Reverse Polish Notation):** Operators come after their operands. - For $(-3 \times -5)$: $-3 -5 \times$ - For $(+3 \times -8)$: $3 -8 \times$ - For $(5 - 3 + (+3 \times -8))$: $5 3 - 3 -8 \times +$ - Division of numerator and denominator: $-3 -5 \times 5 3 - 3 -8 \times + /$ - For $(6 - 2 \times -1 + 3)$: $6 2 -1 \times - 3 +$ - Full expression postfix: $$-3 -5 \times 5 3 - 3 -8 \times + / 6 2 -1 \times - 3 + +$$ 5. **Prefix notation (Polish Notation):** Operators come before their operands. - For $(-3 \times -5)$: $\times -3 -5$ - For $(+3 \times -8)$: $\times 3 -8$ - For $(5 - 3 + (+3 \times -8))$: $+ - 5 3 \times 3 -8$ - Division of numerator and denominator: $/ \times -3 -5 + - 5 3 \times 3 -8$ - For $(6 - 2 \times -1 + 3)$: $+ - 6 \times 2 -1 3$ - Full expression prefix: $$+ / \times -3 -5 + - 5 3 \times 3 -8 + - 6 \times 2 -1 3$$ 6. **Summary:** - Lexicographic (infix): $$(((-3) \times (-5)) / (5 - 3 + ((+3) \times (-8)))) + (6 - (2 \times (-1)) + 3)$$ - Postfix: $$-3 -5 \times 5 3 - 3 -8 \times + / 6 2 -1 \times - 3 + +$$ - Prefix: $$+ / \times -3 -5 + - 5 3 \times 3 -8 + - 6 \times 2 -1 3$$