Problem 30 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$3+4 \cdot 5$$

Step-by-Step Solution

Answer

The simplified result of the expression is 23.

Step 1: Identify the operations

In this expression, we have addition and multiplication: \(3 + 4 \cdot 5\).

Step 2: Follow the order of operations

According to the order of operations, multiplication should be done before addition. Thus, let's multiply 4 by 5 first: \(4 \cdot 5 = 20\). The expression now becomes: \(3 + 20\).

Step 3: Perform the addition

Now, perform the addition operation: \(3 + 20 = 23\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing a mathematical phrase into its simplest form. It's like cleaning up or shortening a sentence while retaining its meaning. To simplify, follow specific rules for combining and reducing numbers or variables.

When simplifying expressions with multiple operations, it's essential to consider the order in which the operations should be done. This helps avoid mistakes. Expressions can include various operations such as addition, subtraction, multiplication, and division.

• Identify all operations involved.
• Use parentheses to group terms if needed.
• Apply the order of operations rules consistently.

Understanding how to simplify expressions is crucial in math, as it lays the foundation for solving more complex problems.

Addition and Multiplication

Addition and multiplication are basic mathematical operations, but they have key differences. Addition combines numbers to yield a total sum. Multiplication, on the other hand, finds the total of one number repeated a specific number of times. This means, when you see expressions involving both, there has to be an order.

• **Addition**: Simply add numbers together, e.g., \(3 + 5 = 8\).
• **Multiplication**: Multiply numbers to find the total, e.g., \(4 \cdot 5 = 20\).

If an expression like \(3 + 4 \cdot 5\) is encountered, multiplication takes priority over addition according to the order of operations. First perform \(4 \cdot 5 = 20\), then add \(3\).

Mathematical Operations

Mathematical operations are the backbone of arithmetic and algebra, including addition, subtraction, multiplication, and division. Together, these operations allow us to solve various mathematical problems. To correctly execute calculations, mathematicians use the **order of operations**. This set of rules dictates the sequence in which operations should be performed:

• Parentheses: Solve anything inside parentheses first.
• Exponents: Evaluate powers and roots.
• Multiplication and Division: From left to right as they appear in the expression.
• Addition and Subtraction: Again, from left to right.

This order is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following these rules ensures everyone gets the same result for a given problem.