Problem 34 - Fundamentals of Mathematics

Question

Rewrite each expression in simpler form. $$ -[-(-7)] $$

Step-by-Step Solution

Answer

The simplified form of \(-[-(-7)]\) is \(-7\).

Step 1: Understanding the Expression

The expression given is \(-[-(-7)]\). This expression contains multiple negative signs which must be dealt with step by step to simplify the expression.

Step 2: Simplify the Inner Negative Sign

Start with the innermost negative sign. Inside the bracket, we have \(-7\). The negative sign just outside the parenthesis will change \(-7\) to \(7\). Thus, \(-(-7) = 7\).

Step 3: Apply the Outer Negative Sign

Now, we simplify \(-[7]\). The negative sign outside the bracket changes \(7\) into \(-7\). Thus the expression \(-[7] = -7\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign (−) in front of the number. Negative numbers are found in various real-world situations such as temperatures below zero or financial debts.
When working with negative numbers, it's important to understand how they interact with each other and with positive numbers. Here’s a quick guide to remember:

• The product or quotient of two negative numbers is positive. For example, (-2) × (-3) = 6.
• The product or quotient of a positive number and a negative number is negative. For example, 2 × (-3) = -6.
• Adding a negative number is the same as subtracting its absolute value: 3 + (-2) = 3 - 2 = 1.
• Subtracting a negative number is the same as adding its absolute value: 3 - (-2) = 3 + 2 = 5.

So, when you encounter negative signs in expressions, remember these rules to help simplify your calculations.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators (like +, –, ×, ÷) that represent a value or relationship. They are a fundamental part of mathematics and appear in many aspects of problem-solving.
To effectively solve problems involving mathematical expressions, you need to understand and apply the order of operations. This usually follows the acronym PEMDAS:

• Parentheses: Do operations inside grouping symbols first.
• Exponents: Evaluate powers and roots.
• Multiplication and Division: From left to right.
• Addition and Subtraction: Also from left to right.

With the expression -[-(-7)], you must consider the sequence where operations within the brackets are simplified first before applying further operations.

Simplification Process

The simplification process involves reducing a mathematical expression to its most basic form, making it easier to understand and work with. The key is to follow the order of operations and systematically apply algebraic rules.
In our example of -[-(-7)], we simplify step by step:

• Step 1: Look at the innermost part of the expression, (-7). The negative sign changes 7 to -7.
• Step 2: Apply the next layer of the negative sign. The expression becomes 7 when you take the negative of -7.
• Step 3: Apply the outermost negative sign to 7, reverting it to -7.

This step-by-step approach ensures you don't miss any operations and achieve the simplest form of the expression efficiently. Simplification not only clarifies the expression but also makes it easier to solve related mathematical problems.