Problem 27 - College Algebra

Question

Simplify the given expression. $$ 12(3-1) \div 6 $$

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Resolve Parentheses

The first operation to perform in the expression is inside the parentheses. Calculate \(3 - 1\). This simplifies to 2. So the expression now is \(12 \times 2 \div 6\).

Step 2: Perform Multiplication

Next, we need to perform the multiplication before division. Multiply \(12 \times 2\) which results in 24. Now the expression becomes \(24 \div 6\).

Step 3: Perform Division

Finally, perform the division. Divide \(24\) by \(6\) to get \(4\). The expression simplifies to 4.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential part of solving mathematical problems. It means rewriting a mathematical expression in its simplest form. The goal is to make calculations easier and clearer by reducing the number of operations involved. In the given exercise, simplifying the expression \[12(3-1) \div 6\] begins by following the Order of Operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Here’s how the simplification process works:

• Look for operations inside parentheses and perform them first.
• Once the parentheses are resolved, address multiplication and division from left to right.
• This step-by-step simplification ensures a structured approach to achieve the final result, in this case, simplifying the entire expression down to 4.

Multiplication and Division

Multiplication and division operations play crucial roles in the order of operations. According to PEMDAS, these operations are to be considered at the same level. Hence, the expression should be solved from left to right, prioritizing neither multiplication nor division over the other.
For the expression \[12 \times 2 \div 6\],this means:

• First, do the multiplication of 12 and 2, which results in 24.
• Then, perform the division of 24 by 6, giving the result of 4.

This sequence ensures that the operations are correctly executed and the expression is simplified accurately. It's important for students to practice this concept to gain confidence in handling more complex equations later on.

Parentheses in Mathematics

Parentheses are used in mathematics to indicate which operations to perform first in an expression. This is a fundamental aspect of the order of operations. When you see parentheses, focus on calculating the operations within them before addressing any other part of the expression.
In the exercise \[12(3-1) \div 6\],parentheses are around \[(3-1)\].

• This indicates you should solve \[(3-1)\],which simplifies to 2, before proceeding with other operations.
• By resolving parentheses first, you eliminate ambiguity and avoid errors that arise from incorrect operation sequences.

Understanding how to handle parentheses corrects ensures accuracy and efficiency in mathematics, helping with consistent results across different types of problems.