Problem 51 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$(-8+5)+(-6+2)$$

Step-by-Step Solution

Answer

The expression simplifies to -7.

Step 1: Identify the Operations

The expression \((-8+5)+(-6+2)\) is composed of addition operations.

Step 2: Solve the First Parenthesis

Focus on simplifying inside the first parenthesis: \(-8 + 5 = -3\).

Step 3: Solve the Second Parenthesis

Next, simplify the second parenthesis: \(-6 + 2 = -4\).

Step 4: Combine the Results

Now sum the simplified results from the parentheses: \(-3 + (-4) = -3 - 4 = -7\).

Key Concepts

Addition Operations

Addition is one of the fundamental operations in mathematics. It is the process of finding the total or sum by combining two or more numbers. In the context of simplifying expressions, particularly those inside parentheses, addition helps us reduce complex expressions step by step.

Consider the expression \((-8 + 5) + (-6 + 2)\). You can break it down into smaller parts using addition operations:

• First, deal with each group within the parentheses separately.
• Simplify the addition within each group.

By doing this, you focus on smaller pieces making the problem more manageable. For example, in the group \(-8 + 5\), you combine these numbers to get \-3\. Similarly, for \(-6 + 2\), you end up with \-4\.This technique helps you systematically work through even longer expressions by tackling each addition operation in turn.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, using different mathematical operations. This makes them easier to work with or evaluate. In our example expression \((-8 + 5) + (-6 + 2)\), we used simplification within each pair of parentheses.

Here is how you simplify expressions step by step:

• Identify and focus on the expressions inside the parentheses first.
• Simplify each part by performing the necessary operations (like addition).
• Combine the results from these operations to find the simplest form of the entire expression.

By simplifying, you reduce the expression \((-8 + 5) + (-6 + 2)\) to \(-3 + (-4)\), which further simplifies to \-3 - 4 = -7\. This process is crucial in helping us understand and solve more complex problems by breaking them down into simpler tasks.

Parentheses in Math

Parentheses are symbols used in mathematics to clearly define the order of operations. When you see an expression like \((-8+5)+(-6+2)\), the parentheses inform you to perform the operations inside them first.

Here's how you use parentheses to manage operations:

• Always resolve operations within parentheses before addressing those outside.
• If there are multiple sets of parentheses, solve them starting from the innermost set.
• Once solved, incorporate their outcomes with the rest of the expression.

In our example, parentheses guide you to first address \(-8+5\) and \(-6+2\). It prevents confusion over which addition should be completed first, ensuring a correct simplification process.Understanding how parentheses dictate the order can significantly influence solving expressions effectively and accurately.