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Chapter 1: Problem 30

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

Short Answer

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

Step by step solution

01

Perform Multiply Operation

According to order of operations, multiply 4 and 5. It gives 20.
02

Perform Add Operation

Next, add 20 and 3 to get 23.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Palo Alto, California, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers.) The mathematical model $$ C=\frac{200 x}{100-x} $$ describes the cost, \(C,\) in tens of thousands of dollars, for removing \(x\) percent of the contaminants. Use this formula to solve. a. Find the cost, in tens of thousands of dollars, for removing \(60 \%\) of the contaminants. b. Find the cost, in tens of thousands of dollars, for removing \(90 \%\) of the contaminants. c. Describe what is happening to the cost of the cleanup as the percentage of contaminants removed increases.

Give an example of an integer that is not a natural number.

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$\frac{5 m-1}{6}=\frac{3 m-2}{4},-4$$

Find this sum, indicated by a question mark. \(3(-3)=(-3)+(-3)+(-3)=?\)

Explain how to determine the sign of a product that involves more than two numbers.
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