Problem 45 - Fundamentals of Mathematics

Question

Determine each value. $$ -(-|-16|) $$

Step-by-Step Solution

Answer

The value is 16.

Step 1: Understand the Problem

We are given the expression \(-(-|-16|)\), and we need to determine its value. This involves understanding the order of operations and the role of negative signs and absolute value.

Step 2: Evaluate the Innermost Expression

Start with the innermost expression within the absolute value symbols. Here, it is \(|-16|\). The absolute value of a number is always positive, so \(|-16| = 16\).

Step 3: Simplify the Expression by Removing Absolute Value

Replace the absolute value expression with its result from Step 2. The expression now becomes \(-(-16)\).

Step 4: Apply the Negative Sign

Evaluate \(-16\), which is simply \(16\) because negating a negative number results in a positive number.

Step 5: Finalize the Expression

Now, evaluate the entire expression. Since applying the negative sign again will return it to a negative number, \(-(-16) = 16\).

Key Concepts

Order of Operations

In mathematics, the order of operations is crucial. It tells us which operations to perform first and which ones come later, especially in complex expressions. The order of operations is like the roadmap that helps us solve problems accurately without getting lost. The standard sequence is abbreviated as PEMDAS:

• Parentheses
• Exponents (or powers and roots, depending on the curriculum)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

For the expression \(-(-|-16|)\), we start by solving the innermost part, which is the absolute value \(|-16|\). This follows the parenthesis rule. Once we simplify this, we can then address the negative signs according to the hierarchical order. By consistently applying these rules, we ensure the accurate simplification of any mathematical problem.

Negative Numbers

Negative numbers represent values less than zero. Working with negative numbers involves paying close attention to signs (positive or negative), ensuring they are correctly applied when performing mathematical operations. Understanding negative numbers is key to solving expressions like \(-(-|-16|)\).

• Positive and Negative Signs: Two negatives make a positive. So in the term \(-(-16)\), the two negatives cancel each other out, changing \(-16\) to \16\.
• Signs in Operations: When multiplying or dividing two negatives, the result is positive. When one is negative, the result is negative.

In essence, dealing with negative numbers in expressions requires careful attention to ensure the correct interpretation of each sign and its effect on the result.

Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators (such as plus and minus) that represent a particular value. To solve these expressions, one must understand the order of operations and the role of each component. The expression \(-(-|-16|)\) involves a few key elements:

• Absolute Value: This function \(|-16|\) provides the distance of the number from zero on a number line, ignoring signs. Here, \|16|\ indicates we should consider the number's absolute quantity.
• Nested Expressions: What happens first depends on what's inside. For \(-(-|-16|)\), the absolute value is resolved first, then the negative signs are applied.

By understanding how to interpret and simplify mathematical expressions correctly, students can solve them accurately, ensuring no steps are overlooked in the simplification process.