Convert the infix expression to prefix expression. Show all steps Infix expression √ A + B * C - D / E + F * (G + H)

To convert an infix expression to a prefix expression, we need to follow a series of steps that involve understanding operator precedence and associativity. The infix expression given is:

√ A + B * C - D / E + F * (G + H)

Here's a step-by-step approach to convert this infix expression to a prefix expression:

Step 1: Parenthesize the expression based on operator precedence

Operator precedence determines the order in which operations are performed. The precedence order from highest to lowest is:

1. Parentheses ()
2. Exponents and roots (like √)
3. Multiplication * and division /
4. Addition + and subtraction -

Based on this precedence, we add parentheses to the expression to make the order of operations explicit:

√(A) + (B * C) - (D / E) + (F * (G + H))

Step 2: Convert to postfix (Reverse Polish Notation) as an intermediate step

To convert to postfix, we use the following rules:

• Operands (variables and constants) appear in the same order as they do in the infix expression.
• Operators are placed after their operands.
• Higher precedence operators appear before lower precedence operators, except when parentheses alter this order.

Let's convert the expression to postfix:

1. Start with the first operator, which is the square root √. In postfix, it comes after its operand: A √

2. Next, we have B * C. In postfix, it becomes: B C *

3. Then, we have D / E. In postfix, it becomes: D E /

4. Finally, we have F * (G + H). We first convert G + H to postfix, which is G H +, and then we append F * to it, which becomes: F G H + *

Now, we combine these parts, respecting the original order and the addition and subtraction operators:

A √ B C * + D E / - F G H + * +

Step 3: Convert from postfix to prefix

To convert from postfix to prefix, we use a stack to reverse the order of operations:

1. Scan the postfix expression from left to right.
2. If the token is an operand, push it onto the stack.
3. If the token is an operator, pop the required number of operands from the stack, combine them with the operator, and push the result back onto the stack.

Let's convert the postfix expression to prefix:

• Start with the first token A. It's an operand, so push it onto the stack.
• Next is the square root operator √. Pop A from the stack, apply the operator, and push the result back onto the stack. The stack now has √A.
• Continue with B and C. Push both onto the stack.
• The next token is *. Pop C and B from the stack, apply the operator, and push the result back onto the stack. The stack now has √A and B*C.
• The next token is +. Pop B*C and √A from the stack, apply the operator, and push the result back onto the stack. The stack now has +√AB*C.
• Continue this process for the rest of the tokens.

The final prefix expression, after processing all tokens, is:

+ - + √A * B C / D E * F + G H

Summary Table

Here's a table summarizing the operator precedence and the steps we took to convert the infix expression to a prefix expression:

Precedence	Operator	Infix Example	Postfix Conversion	Prefix Conversion
1	√	√A	A √	√A
2	* /	B * C, D / E	B C *, D E /	* B C, / D E
3	+ -	A + B, C - D	A B +, C D -	+ A B, - C D
4	( )	(G + H)	G H +	+ G H

Using these rules and the table as a guide, you can convert any infix expression to a prefix expression.