Problem 23 - Algebra A Combined Function

Question

Simplify each expression. $$ 4 \cdot 8-6 \cdot 2 $$

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Identify Operations

The expression contains both multiplication and subtraction. We must deal with these operations according to the order of operations (PEMDAS/BODMAS).

Step 2: Perform Multiplication

Calculate the results for each multiplication operation separately. - Compute the multiplication for the first term: \(4 \cdot 8 = 32\)- Compute the multiplication for the second term: \(6 \cdot 2 = 12\)

Step 3: Perform Subtraction

Now that we have the results from the multiplications, perform the subtraction.Subtract the second product from the first:\(32 - 12 = 20\)

Key Concepts

Multiplication

When working with multiplication in the order of operations, it's crucial to remember that multiplication must be performed before addition or subtraction. This is why multiplication is often referred to as having a higher priority in the order of operations. In our example, the expression given was \(4 \cdot 8 - 6 \cdot 2\). To solve the problem, the first step was to handle each multiplication separately:

• The first multiplication: \(4 \cdot 8 = 32\)
• The second multiplication: \(6 \cdot 2 = 12\)

Breaking the expression into separate multiplication operations allows you to simplify the steps. This leads to a more manageable arithmetic problem. Once each multiplication is resolved, you can confidently proceed to the next operation in the order.

Subtraction

Subtraction is one of the basic operations in mathematics and follows multiplication when applying the order of operations. After completing all necessary multiplications in an expression, you can then proceed to subtraction. In our example problem, once the multiplications were handled, we were left with:

• \(32 - 12\)

Subtraction is simply the process of taking one number away from another. Here, you subtract 12 from 32, which results in 20. It's essential to move through these steps methodically, particularly when an expression includes multiple operations like this one does. Subtraction effectively simplifies the expression further, preparing it for the last simplification step.

Simplification

Simplification is the final phase in solving expressions. The goal of simplification is to reduce a setup that may include various numbers and operations to a single, clear result. It involves applying the rules of arithmetic and the order of operations to successfully boil down the expression to its simplest form. For example, after handling multiplication and subtraction in the given problem, you end up with an expression like \(32 - 12\). Simplifying it means calculating the difference to reach your final answer:

• The simplified result here is 20.

This process highlights the importance of following the correct sequence of operations (PEMDAS/BODMAS). Each step of simplification brings you closer to the final, simplified solution, ensuring clarity and accuracy in problem-solving.