Problem 55 - Prealgebra

Question

Find the value of each expression. $$36 \div(9 \cdot 2)+7$$

Step-by-Step Solution

Answer

The value of the expression is 9.

Step 1: Simplify Inside the Parentheses

Begin by simplifying the expression inside the parentheses. The expression is \(9 \cdot 2\).\ Let’s multiply these two numbers together.\[9 \cdot 2 = 18\]

Step 2: Perform Division

Next, take the result from the parentheses from Step 1 and use it in the division part of the expression. The expression now is \(36 \div 18\).\ Divide 36 by 18.\[36 \div 18 = 2\]

Step 3: Add the Constant

With the division resolved, we move on to the addition part of the expression. The expression at this stage is \(2 + 7\).\ Add 2 and 7 together. \[2 + 7 = 9\]

Key Concepts

Understanding Parentheses in Order of Operations

When working with mathematical expressions, parentheses play a crucial role. They tell you which operation you need to perform first. In our example, the operation inside the parentheses is \(9 \cdot 2\).

Think of parentheses as an indication to "deal with this first." Whenever you see them in a complex expression, focus your attention there. By solving inside the parentheses first, you simplify the rest of the expression.

After calculating \(9 \cdot 2 = 18\), we replace the parentheses in the original expression with 18. This simplification helps set the stage for the next steps.

The Role of Multiplication in Calculations

Multiplication is one of the core operations in mathematics that help in combining numbers in a quick and efficient way. Here, we used multiplication to solve what was inside the parentheses. This is an important step as it helps move the expression closer to a simpler form.

Remember:

• Multiply numbers immediately when they are inside parentheses.
• This operation is performed before both addition and subtraction when not in parentheses.

In our expression example, once you calculated \(9 \cdot 2\), the problem becomes much simpler. Multiplication helped clarify how to progress to the next operations.

Division as a Part of Order of Operations

After dealing with operations inside parentheses and using multiplication, division comes next. Division is essential for breaking larger numbers into smaller parts or groups.

In our expression, once you replace the parentheses with 18, it becomes \(36 \div 18\). Now, divide 36 by 18 to simplify.

Division helps in reducing or segmenting numbers to reach the final value effectively. It is always carried out before addition and subtraction if not included in parentheses. Understanding division's order is key to correct simplification of expressions.

Finally, Addition Completes the Expression

Addition is often one of the final steps in simplifying an expression. It is used to combine numbers to reach a total sum. After reducing the expression through parentheses, multiplication, and division, addition brings us to the answer.

In this case, the expression had become \(2 + 7\). Add these two numbers together to get the final result: 9.

Addition is straightforward, but always ensure that earlier operations are completed to avoid errors. Mastering addition as the final step in order of operations helps complete the problem-solving process smoothly.