Problem 63 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ (0-8)-(4-12) $$

Step-by-Step Solution

Answer

Answer: 0

Step 1: Solve the expressions within the parentheses

In this step, subtract the numbers within the parentheses. (0 - 8) = -8 (4 - 12) = -8 So the expression becomes: $$ (-8) - (-8) $$

Step 2: Perform the subtraction

Now that we have simplified the expressions within the parentheses, we should subtract the results: -8 - (-8) = -8 + 8 = 0 Therefore, the result of the exercise is 0.

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing the expression to its simplest form, making it easier to understand and solve. Simplifying can include combining like terms, applying the distributive property, or, as in our exercise, handling operations inside parentheses.

When you encounter parentheses in algebra, remember they indicate which operations should be completed first. Always perform the operations inside the parentheses as the initial step, before moving on to any outside operations. This follows the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Once the content inside the parentheses is simplified, the rest of the expression becomes clearer, and further operations can be conducted with more ease. Simplifying expressions might not always result in a single number; it could also lead to a simpler expression that can be handled more easily in subsequent steps.

Subtraction with Negative Numbers

Subtraction with negative numbers can often be a source of confusion, but it becomes straightforward with the right approach. First, it's critical to understand what negative numbers represent. A negative number indicates a value less than zero and can be thought of as the opposite of a positive number.

When you subtract a negative number, such as \( -8 - (-8) \), you're actually adding its opposite. This is because subtracting a negative is equivalent to adding a positive—a concept known as the 'additive inverse.' Hence, the expression simplifies to \( -8 + 8 \), which equals 0. This additive inverse property is a fundamental concept in algebra that simplifies the handling of negative numbers.

To ensure understanding, remember this rule: subtracting a negative number is the same as adding its positive counterpart. Visualize a number line if you need to; moving to the left indicates subtraction, while moving to the right indicates addition. So when you subtract a negative, you're essentially moving to the right (adding) the absolute value of that number.

Algebraic Operations

In algebra, operations refer to the basic mathematical procedures we perform on numbers or algebraic expressions. These operations include addition, subtraction, multiplication, division, and sometimes exponentiation. Mastery of these operations is vital because it allows for the manipulation of algebraic expressions to solve equations and inequalities.

Understanding how operations work together, often through the use of the order of operations, enables students to correctly simplify or solve complex expressions. When working with operations in algebra, always consider the 'order of operations' as a strict guide. For example, in our exercise, after simplifying the expressions within the parentheses, the next step was to perform the subtraction operation, following the correct sequence.

Occasionally, you'll see a mix of operations in a more complex expression. In such cases, patience and careful calculation are key. Break the problem down into smaller parts, handling one operation at a time while always adhering to the order of operations, and you'll avoid common mistakes.