Problem 57 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$108+(-456+275)$$

Step-by-Step Solution

Answer

The simplified expression is \(-73\).

Step 1: Identify the Operation Inside the Parentheses

The expression given is \(108 + (-456 + 275)\). According to the order of operations, operations inside parentheses should be performed first. Here, we need to evaluate \(-456 + 275\).

Step 2: Perform the Addition Inside the Parentheses

Evaluate the expression inside the parentheses, \(-456 + 275\). This requires subtracting 275 from 456. \(-456 + 275 = -181\).

Step 3: Add the Result to the Remaining Expression

Now that we have simplified the expression inside the parentheses to \(-181\), the expression becomes \(108 + (-181)\). Add these two numbers together.

Step 4: Perform the Final Addition

Compute the addition: \(108 - 181 = -73\). Thus, the simplification of the expression \(108 + (-456 + 275)\) is \(-73\).

Key Concepts

Parentheses

In mathematical expressions, parentheses are used to indicate which operations should be performed first. This is essential for maintaining the correct order of operations. When you see parentheses in an expression, always address the calculation inside them before moving on to other operations.
This principle comes from the rule known as PEMDAS or BIDMAS, where Parentheses or Brackets come first.
For example, in the expression \(108 + (-456 + 275)\), the operation inside the parentheses is \(-456 + 275\). The order of operations tells us to solve this part first before doing anything outside the parentheses.

• Parentheses help isolate parts of an expression to focus on at a time.
• They ensure that calculations are done in the correct sequence, preventing errors.

In our example, solving \(-456 + 275\) gives the result as \(-181\), simplifying the expression greatly before addressing any remaining calculations.

Addition

Addition is one of the most basic operations in arithmetic and is signified by the plus sign \(+\). In any given mathematical expression, you often need to perform addition to combine numbers or complete calculations.
When dealing with numbers inside parentheses, you begin by handling any addition or subtraction within those boundaries before looking outside.
In our example, after simplifying inside the parentheses, the expression was transformed into \(108 + (-181)\). To execute this addition:

• Recognize \(108\) as a positive number and \(-181\) as a negative number.
• Think in terms of subtraction: essentially, you're finding the difference between \(108\) and \(181\).
• Since \(-181\) is the larger absolute value, the result inherits the negative sign.

By viewing it as subtraction, we can conclude that \(108 + (-181) = -73\). Addition can thus aid in transforming complex problems into simpler, more manageable parts.

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression as simple as possible. This involves combining like terms, performing operations, and following the rules of arithmetic rigorously.
The objective is to arrive at a more digestible final result, without changing the overall value of the expression.
Here are steps to simplify an expression effectively:

• Follow the order of operations (PEMDAS/BIDMAS), addressing parentheses first.
• Perform each operation one step at a time, moving left to right as necessary.
• Combine terms systematically to reduce clutter and identify a clear result.

In our scenario, the original expression \(108 + (-456 + 275)\) was simplified to \(-73\) through orderly operations, all the while reducing potential errors due to correct management of the terms. Simplifying helps in making challenging problems more accessible and the solution process clearer.