Problem 40 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -\\{-[-(-26)]\\} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $26$.

Step 1: Evaluate innermost parentheses

First, we must evaluate the innermost parentheses. In this expression, the only value inside parentheses is -26, so we have: $$-\\{-[-(-26)]\\} = -\\{-[26]\\}$$

Step 2: Evaluate outer parentheses

Now that we have simplified the inner parentheses, we need to evaluate the expression within the outer curly braces. Here, the value inside the braces is multiplying -1 by 26, so we have: $$-\\{-[26]\\} = -\\{-26\\}$$

Step 3: Evaluate final expression

Finally, we must evaluate the remaining expression. In this case, we're negating -26, so we have: $$-\\{-26\\} = 26$$ The simplified version of the given expression is: $$26$$

Key Concepts

Nested Parentheses

Nested parentheses can make expressions look a bit intimidating at first, but they're just a way to group parts of an equation for clarity. When dealing with nested parentheses, the key is to start from the inside and work your way out. This is the same principle used in peeling layers off an onion.

To simplify an expression with nested parentheses, identify which parentheses are the most innermost. In our original exercise, the expression begins with the innermost parentheses around \(-26\), inside square brackets and further within curly braces. The goal is to simplify the innermost expression first, which in our example is simply recognizing that \((-26)\) is already simplified to \(26\).

• Simplify from the innermost set of parentheses first.
• Evaluate the expression step by step, removing parentheses as you go along.

Consider each set of parentheses as a small problem of its own. Once you solve these small problems, the overall equation becomes much easier to manage.

Order of Operations

Order of operations refers to the rules that clarify which procedures should be performed first in a given mathematical expression. Think of it as a universal guideline ensuring that everyone simplifies expressions in the same way, maintaining consistency.

The lack of proper order can lead to confusion and incorrect results. Use the acronym PEMDAS to remember the order:

• Parentheses
• Exponents (or powers)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our original exercise, evaluating the expressions within parentheses came first, following the 'P' in PEMDAS. This strategy ensures that we interpret and simplify mathematical expressions uniformly.

Negative Numbers

Working with negative numbers can sometimes be tricky, especially when combined with nested parentheses.

A negative sign in front of parentheses means to multiply every term inside the parentheses by \(-1\). This is why, in our example, each time we resolved a set of parentheses, we were essentially flipping the sign of the number inside.

• Negative of a negative cancels out to a positive (e.g., \(-(-26) = 26\)).
• Be cautious about signs when simplifying expressions.

Understanding negatives in equations is crucial because it affects the outcome significantly. Flipping a negative inside nested parentheses requires careful attention to ensure accuracy. Ideally, always double-check your sign changes during calculations to avoid errors.