Problem 82 - Algebra for College Students

Question

Simplify each numerical expression. $$-9.3-(10.4+12.8)$$

Step-by-Step Solution

Answer

The simplified expression is -32.5.

Step 1: Understand the Problem

We are tasked with simplifying the expression \(-9.3-(10.4+12.8)\). This requires us to use the order of operations and simplify the expression step-by-step.

Step 2: Simplify the Parentheses

First, calculate the value of the expression inside the parentheses: \(10.4 + 12.8 = 23.2\). Substitute this back into the expression, yielding \(-9.3 - 23.2\).

Step 3: Simplify the Expression

Now, simplify \(-9.3 - 23.2\). Apply the rules for subtracting numbers: Subtracting is the same as adding a negative, so this is equivalent to \(-9.3 + (-23.2) = -32.5\).

Key Concepts

Simplification

Simplification is a fundamental mathematical process that involves breaking down complex expressions into simpler forms. It is the application of the rules of arithmetic to ensure that expressions are easier to handle or evaluate.
Simplifying expressions often involves eliminating parentheses, combining like terms, and reducing fractions or decimals. The main goal is to achieve a more concise representation of the mathematical problem.
For example, in the expression \(-9.3-(10.4+12.8)\), we begin by addressing any operations inside the parentheses, which is a crucial part of following the order of operations. After resolving what's inside the parentheses, the simplified expression becomes easier to manage, allowing us to further reduce the overall complexity and arrive at a straightforward numerical result.

Numerical Expressions

Numerical expressions contain only numbers and operations, without variables. They require careful application of the order of operations to ensure consistency. The order of operations is often remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (or powers)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

Following this order guarantees that numerical expressions are evaluated correctly. For example, in \(-9.3-(10.4+12.8)\), the operation within the parentheses \((10.4 + 12.8)\) is tackled first, leading to \(-9.3 - 23.2\). Finally, subtraction is performed, completing the simplification of the numerical expression.

Negative Numbers

Dealing with negative numbers can be challenging, but mastering their rules is essential for accurate calculations. Negative numbers are numbers less than zero, often used to represent values like debts or drops in temperature.
When simplifying expressions, it’s crucial to understand how to handle operations with negative numbers. Subtracting a positive number is the same as adding its negative counterpart. For example, \(-9.3 - 23.2\) simplifies to \(-9.3 + (-23.2)\). This is a direct application of subtracting as adding a negative.
Ultimately, the calculated result will maintain the sign of whichever number has the greater absolute value. Understanding these rules allows for smooth simplification processes in expressions involving negative numbers.