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Understanding the order of operations is crucial when solving algebraic expressions. This set of rules determines the sequence in which the operations should be performed to correctly solve an expression. Without a universal standard, different people might interpret and solve the same expression in different ways, leading to a myriad of answers. The standard order in which mathematical operations should be carried out is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), often remembered by the acronym PEMDAS or BIDMAS in some countries.

Let's apply this order to the given exercise. Initially, we calculate the expression as it is without parentheses, which adheres to PEMDAS. This initial step is essential as it provides a baseline to compare with once parentheses are introduced. Modifications to the expression by adding parentheses can drastically change the result, which is why testing different placements is necessary to achieve the target value of 45.

Parentheses play a pivotal role in altering the natural order of operations within an algebraic expression. They act as grouping symbols that instruct you to perform the calculations inside them first. This is important to remember as it affects how we manipulate expressions to achieve a desired outcome.

In the solved example, inserting parentheses in different positions changes the terms that are considered a single group. For instance, \(2 \cdot (3+3) \cdot 5\) treats \(3+3\) as a group and calculates it before multiplication. Knowing where to place these parentheses requires an understanding of how each operation interacts and affects the others. Unfortunately, the exercise presented a scenario where no placement of parentheses resulted in the target number, which demonstrates that not all problems can be resolved by simply rearranging groups with parentheses.

BIDMAS and PEMDAS are acronyms that help us remember the order of operations: Brackets/Parentheses, Indices/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). These acronyms reinforce the hierarchy of operations and ensure that we can consistently solve expressions in a logical and systematic way.

In our exercise, we tried to use this hierarchy to achieve a result of 45 by inserting parentheses, which affects the overall outcome. However, after following BIDMAS/PEMDAS and testing all possible parenthesis placements, it was found that no combination produced the target value. This exploration process is a practical application of the BIDMAS/PEMDAS rule, emphasizing its significance in guiding algebraic problem-solving.

Algebraic problem-solving is a step-by-step approach that involves understanding the problem, devising a plan, carrying out the plan, and then looking back to check and interpret the results. In this context, each step is carefully designed to use algebraic rules such as the order of operations and the proper use of parentheses to find a solution.

In the provided example, the problem-solving process involved experimenting with different placements of parentheses—the plan—to reach the desired outcome of 45. The final look back step confirmed that with the given numbers and operations, the exercise could not be solved as intended. This iterative approach of trial and error is central to algebraic problem-solving and develops analytical skills that are used to explore all possible avenues for solving an expression.