Problem 38 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -[-(-6)] $$

Step-by-Step Solution

Answer

Answer: -6

Step 1: Work with the Innermost Negatives

The given expression is $$ -[-(-6)] $$ First, let's perform the operation inside the innermost parenthesis: $$ -(-6) $$ We know that the product of two negatives is a positive, so this simplifies to: $$ 6 $$

Step 2: Work with the Outermost Negatives

Now, we will perform the operation outside: $$ -(6) $$ Here we have a negative sign outside the parenthesis, which means the result will be negative. $$ -6 $$

Step 3: Final Answer

After simplifying and working through the negative signs, the final answer is: $$ -6 $$

Key Concepts

Negative Numbers

Understanding negative numbers is crucial when dealing with arithmetic operations, especially those involving subtraction and multiplication. Negative numbers are numbers less than zero and are denoted with a minus sign (-) in front of them. For instance, -1, -20, and -100 are all negative numbers. They represent values that are less than zero and are commonly used to signify opposites or debts in real life, like when dealing with temperature below freezing or financial deficits.
Using negative numbers in arithmetic can be tricky if you're not familiar with the rules, but here's a quick rundown:

• Adding two negative numbers results in a more negative number.
• Subtracting a negative number is the same as adding its positive counterpart (e.g., \(-(-6) = 6\)).
• The product or division of two negative numbers results in a positive number.
• Multiplying or dividing a positive number by a negative one results in a negative product.

Understanding these basic principles will help simplify negative number operations.

Parentheses in Math

Parentheses play a vital role in mathematical expressions. They dictate the order of operations by indicating which operations to perform first. Parentheses are like instruction brackets, guiding us in solving expressions by highlighting operations that need immediate attention.
When you see an expression wrapped in parentheses, it should be solved before performing other operations outside. This is part of the Order of Operations, usually remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• First solve operations inside parentheses.
• Then move to any exponents related to the expression.
• Multiplication and division come next.
• Addition and subtraction come last.

For example, in the expression \(-[-(-6)]\), we first solve the innermost operation \(-6\) within the parentheses, and then evaluate from the inside out, as the parentheses instruct.

Simplifying Expressions

Simplifying expressions is a fundamental skill in math that involves reducing expressions to their simplest form while retaining their value. This process helps you see the underlying structure of an equation or expression, making it easier to solve or understand.
The process typically involves a series of steps:

• First, address any operations inside parentheses. This includes handling negative signs, as in simplifying \(-(-6)\) to 6.
• Next, apply the rules of arithmetic, such as adding, subtracting, multiplying, or dividing terms.
• Combine like terms if applicable to further reduce the expression.

Using our example, \(-[-(-6)]\), the goal is to systematically eliminate the nested parentheses by addressing each layer one at a time, starting from the innermost to the outermost. End with an answer that is clean and free of unnecessary complexity. In this case, the expression simplifies down to -6.