Problem 43 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -\\{-[-(-31)]\\} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is 31.

Step 1: Identify the innermost expression

The inner expression we need to simplify is "\(-(-31)\)".

Step 2: Simplify the negative of negative

In order to remove the double negative, keep in mind that minus of a negative number is equal to a positive number. So, \(-(-31) = 31\).

Step 3: Substitute the simplified innermost expression

Now we have the following simplified expression: $$ -\\{-[31]\\} $$

Step 4: Identify the next innermost expression

The next inner expression is "\(-{[31]}\)".

Step 5: Simplify the brackets

Remove the square brackets and the minus sign. So, \(-{[31]} = -31\).

Step 6: Substitute the simplified expression

Now we have the following simplified expression: $$ -\\{-31\\} $$

Step 7: Identify the final innermost expression

The final inner expression is "\(-\\{-31\\}\)".

Step 8: Simplify the negative of negative

In order to remove the double negative, keep in mind that minus of a negative number is equal to a positive number. So, \(-\\{-31\\} = 31\). The final simplified expression is: $$ 31 $$

Key Concepts

Negative Signs Simplification

Understanding how negative signs work in mathematics can be really handy when simplifying complex expressions. When you see multiple negative signs together, this is called a double negative. A double negative can be simplified to a positive. This is because negating a negative number results in a positive number. For example,

• \(-(-31)\) becomes \(31\).

Think of it like walking backwards twice: if you start facing a direction, turn around once (going backward), and then turn around again, you'll end up facing your original direction. That's similar to what happens with double negatives in math. Sometimes, you may encounter expressions with several layers of negative signs, like in nested brackets or parentheses. Just remember to simplify one level at a time, starting from the innermost expression and working outwards, effectively untangling the negative signs as you go. This will ensure your solution path is both smooth and logical.

Order of Operations

When tackling complex mathematical expressions, it's essential to follow the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This helps us solve expressions accurately by processing operations in the correct sequence. Let's break this down: when you see parentheses, it's a signal to resolve the contents first. For example, in the expression \(-{[-31]}\), tackle what's inside the brackets first; simplifying this is your top priority.

• Evaluate innermost operations first, like \(-(-31) = 31 \).
• Move outward, simplifying as you progress, like \([-31]\) next, and so forth.

Ensuring the order of operations is adhered to prevents mistakes and guarantees accurate outcomes, especially in algebraic expressions where brackets and negative signs coexist. A structured approach using PEMDAS provides clarity while simplifying or solving more complicated algebraic equations.

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. It's a language of its own, used to solve problems by setting up equations to represent real-world situations or abstract scenarios. In algebra, simplifying expressions using negative signs or brackets is common practice. Let's look at the expression \(-\{-[-(-31)]\}\). Algebraic rules guide us in breaking this down:

• Recognize any pattern of negative signs and apply the rules of negative simplification.
• Simplify step by step, handling each level of nested operations individually.

Algebra often requires applying logical reasoning and established mathematical rules to find a solution. Understanding the fundamental principles of algebra can simplify expressions and provide a pathway to handling even the most complex equations systematically. Remember, the key in algebra is to approach problems methodically, simplifying components before tackling the bigger picture.