Convert the infix expression ((A + B) * C - (D - E) ^ (F + G)) into the equivalent postfix notation.

Introduction / Context:
Converting expressions between infix, postfix, and prefix notation is a classic data structures and algorithms topic. It tests your understanding of operator precedence, associativity, and the use of stacks. Infix notation is the usual human readable form, while postfix notation, also known as Reverse Polish notation, is often used for evaluation by stack based machines or interpreters. This question asks you to convert a moderately complex infix expression with parentheses, multiplication, subtraction, and exponentiation into postfix form.

Given Data / Assumptions:

• Infix expression: ((A + B) * C - (D - E) ^ (F + G)).
• Operators involved: addition, subtraction, multiplication, and exponentiation.
• The usual precedence order applies: exponentiation highest, then multiplication, then addition and subtraction.
• Parentheses must be respected in the conversion process.

Concept / Approach:
To convert an infix expression to postfix, you can use a stack based algorithm. Operands are output directly, while operators are pushed to a stack based on precedence and associativity. Parentheses control when operators are popped from the stack. Another helpful approach is to convert subexpressions one by one, starting with the innermost parentheses, while always respecting operator precedence. The final postfix string should represent the same computation order as the original infix expression.

Step-by-Step Solution:
Step 1: Consider the subexpression (A + B). Its postfix form is AB +. Step 2: Multiply this result by C as in (A + B) * C. The postfix form becomes AB + C *. Step 3: Consider the subexpression (D - E). Its postfix form is DE -. Step 4: Consider the subexpression (F + G). Its postfix form is FG +. Step 5: Now apply exponentiation to (D - E) ^ (F + G). In postfix notation, the operands come first followed by the operator, so the result is DE - FG + ^. Step 6: The outer expression subtracts this exponentiation result from (A + B) * C. We already have AB + C * for the left side and DE - FG + ^ for the right side. Subtraction in postfix puts the operator after both operands, so the full postfix expression is AB + C * DE - FG + ^ -.

Verification / Alternative check:
As a check, count the operands and operators. The original expression has operands A, B, C, D, E, F, G and operators plus, minus, times, minus, plus, and exponentiation, followed by an outer minus, for a total of seven operands and six operators. The postfix expression AB + C * DE - FG + ^ - contains seven operand symbols and six operator symbols, which matches. The structure also preserves the correct precedence because exponentiation is applied directly to DE - and FG + before the final subtraction.

Why Other Options Are Wrong:
Option A: Starts with AC instead of AB and misplaces operands, so it does not correspond to the original expression. Option C: Uses AC and changes the order of operations for the second half, leading to an incorrect computation. Option D: Uses AB correctly at the start but alters the order of operations in the exponent and subtraction by using -+ instead of +^ in the correct positions.

Common Pitfalls:
A common mistake is to forget operator precedence and simply process the expression from left to right. Another pitfall is to incorrectly handle parentheses, especially when exponentiation is involved. Students also sometimes misplace the minus operator at the end, which changes the meaning of the expression. Practising with a stack based algorithm and verifying against the order of operations helps avoid these errors.

Final Answer:
The correct postfix notation for the given infix expression is AB + C * DE - FG + ^-.