Problem 74 - Intermediate Algebra

Question

Simplify each numerical expression. $$ 7(8-9)+(-6)(4) $$

Step-by-Step Solution

Answer

The simplified expression is -31.

Step 1: Simplify the Parentheses

First, simplify the expression inside the first parentheses: \(8 - 9 = -1\). Now the expression is: \(7(-1) + (-6)(4)\).

Step 2: Multiply Numbers

Multiply the numbers from the simplified expression:- Calculate \(7 \times (-1) = -7\).- Calculate \((-6) \times (4) = -24\).Now the expression becomes: \(-7 + (-24)\).

Step 3: Add the Results

Add the results from the step 2:- Calculate \(-7 + (-24) = -31\). Thus, the simplified expression is \(-31\).

Key Concepts

Numerical Expression

A numerical expression is an important way to represent mathematical situations using numbers and operations. It gives you a collection of numbers, operations, and sometimes variables, coming together to make one big calculation. For example, the expression \( 7(8-9)+(-6)(4) \) is a numerical expression.

These expressions can have multiple parts which need to be simplified in a certain order, making it essential to understand each component. Understanding these expressions properly can help in performing the right calculations.

In dealing with numerical expressions:

• Recognize the numbers and how they're combined.
• Identify which arithmetic operations are applied.
• Follow a systematic approach to simplify them.

Breaking down the expression properly and simplifying them in steps allows you to solve them easily and accurately.

Arithmetic Operations

Arithmetic operations are the basic processes through which we interact with numbers. The main operations include addition (+), subtraction (-), multiplication (×), and division (÷). In a numerical expression like \( 7(8-9)+(-6)(4) \), understanding each operation is vital.

Here's how you can approach these operations:

• **Addition**: Combining numbers to increase the value.
• **Subtraction**: Removing or decreasing the numbers from one another.
• **Multiplication**: Repeatedly adding a number, which can also affect the sign based on the inputs.
• **Division**: Splitting a number into equal parts or groups.

When simplifying, especially in complex expressions that mix these operations, adhering to the correct order of operations is crucial to reach the correct solution.

Parentheses

Parentheses play a fundamental role in numerical expressions by marking out which operations should be done first. In \( 7(8-9)+(-6)(4) \), the parentheses tell you what to focus on before anything else. Whenever you see them in an equation, it’s a signal to simplify that part first.

• Typically, you'll see operations like addition or subtraction inside the parentheses.
• Always solve what's inside the parentheses to get the correct outcome.
• If more than one operation is present, keep the order of operations in mind.

Remember this basic rule: **Operations inside the parentheses take priority over others**. Working inside the parentheses first ensures that the overall expression simplifies correctly, maintaining the relationship between numbers and operations.