1. **Stating the problem:** We want to write the expression $$\frac{-3 \times -5}{5 - 3 + (+3 \times -8)} + (6 - 2 \times -1 + 3)$$ in lexicographic (infix), postfix, and prefix notations. 2. **Expression given:** $$\frac{-3 \times -5}{5 - 3 + (+3 \times -8)} + (6 - 2 \times -1 + 3)$$ 3. **Recall notation definitions:** - **Lexicographic (infix):** Operators are written between operands, e.g., $a + b$. - **Postfix (Reverse Polish Notation):** Operators come after their operands, e.g., $a b +$. - **Prefix (Polish Notation):** Operators come before their operands, e.g., $+ a b$. 4. **Break down the expression into parts:** - Numerator: $-3 \times -5$ - Denominator: $5 - 3 + (+3 \times -8)$ - Second term: $6 - 2 \times -1 + 3$ 5. **Lexicographic (infix) form:** $$\frac{-3 \times -5}{5 - 3 + (+3 \times -8)} + (6 - 2 \times -1 + 3)$$ 6. **Postfix conversion:** - Numerator: $-3 -5 \times$ - Denominator: $5 3 - 3 -8 \times + +$ - Fraction: $-3 -5 \times 5 3 - 3 -8 \times + + /$ - Second term: $6 2 -1 \times - 3 +$ - Whole expression: $-3 -5 \times 5 3 - 3 -8 \times + + / 6 2 -1 \times - 3 + +$ 7. **Prefix conversion:** - Numerator: $\times -3 -5$ - Denominator: $+ - 5 3 \times 3 -8$ - Fraction: $/ \times -3 -5 + - 5 3 \times 3 -8$ - Second term: $+ - 6 \times 2 -1 3$ - Whole expression: $+ / \times -3 -5 + - 5 3 \times 3 -8 + - 6 \times 2 -1 3$ **Final answers:** - Lexicographic (infix): $$\frac{-3 \times -5}{5 - 3 + (+3 \times -8)} + (6 - 2 \times -1 + 3)$$ - Postfix: $$-3 -5 \times 5 3 - 3 -8 \times + + / 6 2 -1 \times - 3 + +$$ - Prefix: $$+ / \times -3 -5 + - 5 3 \times 3 -8 + - 6 \times 2 -1 3$$