Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$7+6 \cdot 3$$

Step-by-Step Solution

Answer

After applying the order of operations to the given expression, the simplification result is 25.

Step 1: Identifying Operations

Identify the operations present in the expression. In this case, we have an addition operation \(7 +\) and a multiplication operation \(6 \cdot 3\). According to the order of operations, multiplication should be performed before addition.

Step 2: Performing Multiplication

Multiply 6 by 3 to get 18. And the expression becomes \(7 + 18\).

Step 3: Performing Addition

In the final step, add 7 to 18 to end up with 25.

Key Concepts

Simplifying Expressions

When working on mathematical problems, simplifying expressions is a fundamental skill that students need to master. It involves reducing an expression to its simplest form while ensuring it maintains its original value. Simplifying may involve combining like terms, eliminating parentheses, and following specific orders for carrying out arithmetic operations.

For example, given the expression from our exercise, the initial form of the expression is not immediately revealing of its simplest numeric equivalent. The process of simplifying will transform the expression from a combination of numbers and operations into a single numerical value. As we saw in the provided solution, through simplification, the compound expression \(7 + 6 \cdot 3\) was efficiently reduced to its simplest form of \(25\). This step-by-step approach not only gives the final answer but also allows students to understand the structure and the transformations that the original expression undergoes.

Arithmetic Operations

The foundations of arithmetic involve several basic mathematical operations: addition, subtraction, multiplication, and division. These are the tools we utilize to approach a broad range of problems.

In our example, the expression \(7 + 6 \cdot 3\) comprises two different operations:

• Addition (the \(+\) symbol indicates that two numbers will be added together)
• Multiplication (the \(\cdot\) symbol denotes the times operation, where one number is taken as many times as the value of the other number)

Understanding how to correctly apply these operations is essential. Each operation follows specific rules that govern how we combine numbers to reach a solution. Particularly, knowing when each operation needs to be performed is vital—a concept encapsulated by the acronym PEMDAS, explained further in the section on the order of operations.

PEMDAS

The acronym PEMDAS is a mnemonic that helps students remember the order in which they should perform operations when simplifying expressions. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In adhering to PEMDAS for our exercise, you'll notice:

• There are no parentheses or exponents, so we move on to multiplication or division.
• The expression contains a multiplication operation, which we perform first before addition, resulting in \(7 + 18\).
• Lastly, we handle the addition, yielding the final simplified result of \(25\).

By following PEMDAS, students can systematically approach and accurately simplify complex numerical expressions. It is a universally recognized guideline that ensures consistency and correctness in the way arithmetic problems are solved.