Problem 74 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ -(-7 m) $$

Step-by-Step Solution

Answer

The simplified form of \(-(-7m)\) is \(7m\).

Step 1: Identify the Problem

We need to simplify the expression \(-(-7m)\). Let's understand that we're dealing with a negative sign applied to another negative term.

Step 2: Apply the Rule of Signs

The expression \(-(-7m)\) has two negative signs. According to the rule of signs, when two negatives are adjacent, they cancel out to become a positive sign. So, the expression turns into \(7m\).

Step 3: Rewrite with Positive Sign

After cancelling out the negative signs, the simplified form of the expression \(-(-7m)\) is \(7m\).

Key Concepts

Understanding Negative Numbers

Negative numbers are a fundamental concept in mathematics, representing values less than zero. They are often used to describe losses, depths below sea level, or temperatures below freezing. When you see a negative number, it is typically indicated by a minus sign (-) in front of the number, such as -7.

A negative number is not just a lower magnitude, but it's inherently different in nature, representing an opposite direction or effect from positive numbers.
Consider the temperature: if zero represents freezing, then -7 could indicate very cold weather. Understanding negative numbers sets the foundation for various algebraic operations, including simplifying expressions that contain them.

The Rule of Signs

The rule of signs is a handy set of guidelines that helps simplify expressions involving both positive and negative numbers.

In mathematics, these rules dictate how the product or quotient of two signed numbers operates:

• Two positive numbers multiply or divide to give a positive result.
• Two negative numbers multiply or divide to give a positive result.
• A positive and a negative number multiply or divide to give a negative result.

When simplifying expressions like -(-7m), this rule becomes essential. The two negative signs cancel each other out, leading us to a positive value, emphasizing the importance of understanding and applying these rules correctly in algebraic settings.

Expression Simplification

Expression simplification in algebra involves reducing expressions to their simplest form for easier comprehension and computation. The aim is to make them as straightforward as possible while retaining their original value. Simplification often includes removing brackets, combining like terms, and resolving operations between numbers, particularly concerning the rule of signs.


To simplify an expression such as -(-7m):

• First, recognize any pairs of negative signs that can be canceled out, as they turn positive.
• Reevaluate the expression after applying these operatiors to see any further possible reductions.
• In this example, you end up with 7m, a much cleaner and concise version of the original expression.

Expression simplification is crucial in keeping calculations efficient and revealing the underlying simplicity of seemingly complex expressions.