## Infix to Prefix Conversion: (a+b^c)*d+e^5

Infix notation is the standard mathematical notation we use, where operators are placed **between** operands. Prefix notation, also known as Polish notation, places operators **before** their operands. Converting from infix to prefix can be useful for evaluating expressions efficiently using a stack-based approach.

Here's how we can convert the expression **(a+b^c) d+e^5* from infix to prefix:

### 1. Shunting-Yard Algorithm

The Shunting-Yard Algorithm is a popular method for converting infix expressions to postfix (RPN). We can adapt this algorithm for prefix conversion.

**Steps:**

**Initialization:**Create an empty output queue and an empty operator stack.**Process the expression from left to right:****If you encounter an operand (a, b, c, d, e):**Add it to the output queue.- *
*If you encounter an operator (+,**, ^):***While the operator stack is not empty and the top operator has higher precedence than the current operator (or the same precedence and is left-associative):**Pop the operator from the stack and add it to the output queue.**Push the current operator onto the stack.**

**If you encounter a left parenthesis:**Push it onto the stack.**If you encounter a right parenthesis:****Pop operators from the stack and add them to the output queue until you encounter a left parenthesis.****Remove the left parenthesis from the stack.**

**After processing the expression:**Pop any remaining operators from the stack and add them to the output queue.

### 2. Applying the Algorithm

Let's apply the algorithm to our expression:

```
(a+b^c)*d+e^5
```

**Output Queue:**Empty**Operator Stack:**Empty**Process (:**Push '(' onto the stack.**Process a:**Add 'a' to the output queue.**Process +:**Push '+' onto the stack.**Process b:**Add 'b' to the output queue.**Process ^:**Push '^' onto the stack.**Process c:**Add 'c' to the output queue.**Process ):**Pop '^' and '+' from the stack and add them to the output queue. Remove '(' from the stack.- **Process
*: ** Push '*' onto the stack. **Process d:**Add 'd' to the output queue.**Process +:**Push '+' onto the stack.**Process e:**Add 'e' to the output queue.**Process ^:**Push '^' onto the stack.**Process 5:**Add '5' to the output queue.**Process End:**Pop '^' and '+' from the stack and add them to the output queue.

### 3. Result

The final output queue contains: **+ * + a ^ b c d ^ e 5**. This is the prefix notation of the expression.

### 4. Evaluation

To evaluate the prefix expression, we would process it from right to left, performing operations as we encounter them. This would involve using a stack for storing operands and intermediate results.

This process demonstrates how the Shunting-Yard Algorithm can be adapted to efficiently convert infix expressions to prefix, opening the door for stack-based evaluation and other computational benefits.