Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$3+4 \cdot 5$$

Step-by-Step Solution

Answer

The simplified expression of \(3+4 \cdot 5\) is 23.

Step 1: Perform Multiply Operation

According to order of operations, multiply 4 and 5. It gives 20.

Step 2: Perform Add Operation

Next, add 20 and 3 to get 23.

Key Concepts

Multiplication

When delving into the world of order of operations, multiplication takes precedence over addition. This order is critical because it ensures that mathematical expressions are evaluated in a consistent, correct manner.
In our example expression, "\(3 + 4 \cdot 5\)", multiplication comes first.

• First, identify the numbers and the multiplication operation: 4 and 5.
• Multiply 4 and 5 to get 20.

This operation simplifies part of the expression, reducing it from three terms to just two terms. It's akin to transforming a more complex puzzle into a simpler one. Remember: performing multiplication before addition in this setting is not arbitrary but rather a rule designed for clarity and consistency in mathematics.

Addition

After multiplying in the given expression, the next task is handling addition. This is the step where we combine terms together to further simplify the expression.

Returning to our expression "\(3 + 4 \cdot 5\)", once we have simplified the multiplication to 20, we can focus on the addition:

• The expression is now "\(3 + 20\)".
• We simply add 3 and 20 together.
• The result is 23.

Addition is straightforward once previous operations (like multiplication) have been properly addressed. It's about bringing all your simplified results together into one final answer. This demonstrates how understanding and applying the correct order of operations leads to finding the correct solution.

Simplification

Simplification is the goal of applying the order of operations, transforming a complex expression into its simplest form. By addressing each component, the given mathematical expression becomes more manageable.

Initially, the expression may seem daunting, but by taking steps to simplify, we make it understandable:

• Start by applying the order of operations: multiplication before addition.
• This results in simplified intermediate expressions: first from "\(4 \cdot 5\)" to 20, then reducing "\(3 + 20\)" to 23.

Remember, the essence of simplification is reducing confusion and error by dealing with the components of the expression in the right order. Observing and following these steps not only gets us to the correct solution but also aids in grasping the intricacies of other mathematical problems.