Problem 120 - Elementary Algebra

Question

Simplify. $$ -|-(-21)| $$

Step-by-Step Solution

Answer

The expression simplifies to -21.

Step 1: Understand the Expression

The expression we need to simplify is \(-|-(-21)|\). It involves nested absolute values and a negative sign.

Step 2: Simplify the Innermost Absolute Value

First, we find the absolute value of the innermost expression. The innermost expression is \(-(-21)\). Since subtracting a negative number is the same as adding the positive, this becomes 21. So, it simplifies to:\(|21|\).

Step 3: Evaluate Absolute Value

Next, we take the absolute value of 21. The absolute value of any positive number is the number itself, so \(|21| = 21\).

Step 4: Apply the Outer Negative

Finally, we apply the outer negative sign to the result found in Step 3. The expression \(-|21|\) simplifies to \(-21\).

Key Concepts

Nested Absolute Values

When dealing with nested absolute values, it is important to understand the concept of layers. An "absolute value" is always non-negative and indicates the distance from zero on a number line. In this exercise, we encounter nested absolute values in the expression \(-|-(-21)|\).

To simplify such expressions, focus on solving the innermost part first. This involves breaking down the expression layer by layer:

• Identify the innermost expression within the absolute value brackets.
• Apply the absolute value operation to it step-by-step.
• Resolve the outer absolute value once the inner expression has been simplified.

By working from the inside out, the nested operations often become much easier to manage and understand. The key is to simplify each level before moving outward to the next one.

Negative Numbers

In mathematics, negative numbers can sometimes cause confusion, especially when combined with absolute value operations. The negative sign simply indicates a direction on the number line opposite to positive numbers. In this problem, the negative sign is used in different roles:

• Reversing the sign of a positive number, e.g., \(-(-21)\) becomes 21.
• Indicating the opposite of the whole expression, e.g., applying \(-|21|\) leading to \(-21\).

Understanding the influence of these negative signs is crucial. Subtracting a negative is equivalent to adding a positive and flipping the sign of an expression is commonly needed when simplifying.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. This makes problems easier to understand and solve. In this case, we simplify the expression \(-|-(-21)|\) by systematically applying rules for absolute values and negative numbers.

Here are the steps:

• Evaluate the innermost operation: Process \(-(-21)\) to become 21.
• Apply the absolute value: Since \(21\) is positive, \(|21|\) remains 21.
• Manage extreme expressions like \(-|21|\), changing it to \(-21\).

Following these steps ensures the expression is simplified efficiently. Always address parentheses first, then absolute values, and finally apply any negative operations as they appear.