Problem 92 - Elementary Algebra Within Reach

Question

In Exercises 89-96, evaluate the expression. $$ -(-8)+6-4-2 $$

Step-by-Step Solution

Answer

The expression '-(-8) + 6 - 4 - 2' equals '8'

Step 1: Understand the Negation of a Negative Number

The first operation to perform is the '-(-8)'. The negation of a negative number is a positive number. Therefore, '-(-8)' is equal to '+8'.

Step 2: Perform the Other Operations

Now perform the remaining operations in the sequence they appear from left to right. So, replacing '-(-8)' with '+8', the expression becomes: '+8 + 6 - 4 - 2'.

Step 3: Final Calculation

When you add and subtract the remaining numbers, '+8 + 6 - 4 - 2', you get '14 - 4 - 2' which is '10 - 2', finally you get '8'.

Key Concepts

Negation of Numbers

The concept of negation in mathematics plays a crucial role when evaluating expressions involving negative numbers. When you see a negative sign in front of a number, it indicates the opposite of that number. However, when a negative sign appears in front of another negative sign, it transforms the value into a positive number. This is because negating a negative is essentially reversing the direction twice, making it positive.

For example, if we take the expression \(-(-8)\), it translates to the opposite of \(-8\), which is \(+8\). It's similar to removing two negatives in everyday language: if it's not 'not fun', then it is simply 'fun'.

Understanding this concept is key when tackling problems involving multiple negative numbers. Always remember that two negatives make a positive. This knowledge not only simplifies calculations but also helps prevent errors in arithmetic operations.

Arithmetic Operations

Arithmetic operations are fundamental to algebra, ensuring we can solve expressions and equations correctly. The basic operations include addition, subtraction, multiplication, and division. In the given expression, \(-(-8) + 6 - 4 - 2\), several arithmetic operations are at play. Here’s a breakdown of how to handle them:

• **Addition**: Combining numbers to get a sum. In \(+8 + 6\), both numbers are added directly.
• **Subtraction**: Removing a number from another. We can see this in \(14 - 4\) and \(10 - 2\).

Handling arithmetic operations step-by-step ensures accuracy as each operation changes the numbers involved. Try solving each operation one at a time, rather than combining too many steps in one go. Simplifying expressions using correct arithmetic operations allows you to reach the correct answer with ease. Understanding these basic operations is critical for all further mathematical learning.

Order of Operations

When evaluating expressions, it is essential to follow the order of operations. This ensures that we tackle each part of an expression in the correct sequence, leading to the right answer. A common way to remember the order of operations is by the acronym PEMDAS:

• **P**: Parentheses
• **E**: Exponents
• **MD**: Multiplication and Division (from left to right)
• **AS**: Addition and Subtraction (from left to right)

In the expression \(-(-8) + 6 - 4 - 2\), we first handle the negation, as stated earlier, turning \(-(-8)\) into \(+8\). After applying the negation, there are no parentheses or exponents, so we move straight to addition and subtraction.

Always perform these operations from left to right. Start by calculating \(+8 + 6\) to get \(14\), then \(14 - 4\) to get \(10\), and finally \(10 - 2\) to arrive at \(8\). This methodical approach ensures that no steps are missed, and each part of the expression is correctly or logically addressed, resulting in the accurate final answer.