Problem 32 - Fundamentals of Mathematics

Question

For the following 6 problems, rewrite each number in simpler form. $$ -[-(-8)] $$

Step-by-Step Solution

Answer

The simplified form is -8.

Step 1: Understanding the Problem

The expression we need to simplify is \(-[-(-8)]\). This is a number with several nested negative signs, and the goal is to simplify the number by removing the unnecessary negatives.

Step 2: Inner Parentheses

First, consider the innermost expression \(-8\). This is just the number -8.

Step 3: Evaluate Double Negative

Next, look at \(-(-8)\). According to the rules of arithmetic, a double negative cancels out, making the expression positive. Therefore, \(-(-8)\) simplifies to 8.

Step 4: Outer Parentheses

Finally, evaluate the entire expression \(-[8]\). This is equivalent to just \(-8\) since there is a negative sign outside of the 8.

Key Concepts

Negative Numbers

Negative numbers can often seem a bit confusing, especially when you have multiple negative signs like in the problem \(-[-(-8)]\). It's important to remember that a negative number is simply any number less than zero. Such numbers are typically written with a minus sign in front, like \(-8\), which means 8 units below zero.
When you encounter multiple negative signs, think about them in pairs. Each pair of negative signs essentially cancels out, similar to saying 'not not doing something' means you are doing it. For example, in the expression \(-(-8)\), the two negatives cancel each other out, converting the negative into a positive, resulting in \(8\).

• One negative sign before a positive number turns it into a negative.
• Two negative signs in a row turn the value into a positive.
• Three negative signs make the expression negative again.

Arithmetic Rules

Understanding basic arithmetic rules is key to simplifying expressions with negative numbers. In math, arithmetic rules dictate how numbers interact in operations like addition, subtraction, multiplication, and division.
One of the rules is about negative signs: when you multiply or divide two negative numbers, their negatives cancel out, resulting in a positive number. However, adding or subtracting with negatives is different. It's more about the direction on the number line. If you subtract a negative number, you're essentially adding its positive counterpart.

• Multiplication or division of two negatives: the result is a positive.
• Adding two negatives: the sum is more negative or further from zero.
• Subtracting a negative: effectively the same as adding a positive.

Learning these fundamental rules helps in understanding why expressions simplify the way they do, for instance why \(-(-8)\) becomes \(8\).

Parentheses Evaluation

Evaluating parentheses correctly is crucial in simplifying complex expressions. Parentheses in math are used to denote operations that need to be performed first. Following the order of operations, operations inside parentheses have priority.
Starting with the innermost parentheses, you solve or simplify the expression, and work your way out. This approach prevents mistakes and ensures you tackle the problem systematically.
In the given expression \(-[-(-8)]\), we first evaluate the innermost part, which is \(-8\).

• Begin with the deepest level of parentheses and solve outward.
• Apply any arithmetic rules necessary to simplify the expression inside parentheses.
• Continue until all parentheses are evaluated, simplifying at each step.

Once \(-(-8)\) is simplified to \(8\), removing the inner parentheses, we then add the outer negative sign to result in \(-8\).