Problem 26 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(3+8 \cdot(6-2)+11\)

Step-by-Step Solution

Answer

The value of the expression is 46.

Step 1: Parentheses first

According to the order of operations, we first solve the expression inside the parentheses: \(6 - 2 = 4\). The expression now becomes: \(3 + 8 \cdot 4 + 11\).

Step 2: Multiplication

Next, we perform multiplication. Multiply 8 by the result of the parentheses:\(8 \cdot 4 = 32\).The expression now becomes: \(3 + 32 + 11\).

Step 3: Addition

Finally, we perform addition from left to right: \(3 + 32 = 35\) and \(35 + 11 = 46\).

Key Concepts

Parentheses: The First Step in Order of Operations

When dealing with mathematical expressions, it is important to follow the order of operations. The first step is to tackle any operations within parentheses. Parentheses serve to group parts of an expression, indicating that this should be solved first. This is crucial because it can change the outcome of the entire calculation.

For example, in our exercise, we have the expression \(6 - 2\) inside parentheses. By solving it first, we simplify it to \(4\). This step ensures that you handle complex expressions correctly and maintain the proper calculation flow.

• If a problem has nested parentheses, always start with the innermost pair.
• Once all operations inside parentheses are completed, move on to the next steps following the order of operations.

Multiplication: A Precedent over Addition

After solving expressions within parentheses, the next step in the order of operations is to perform multiplication or division. In our exercise, after replacing \((6-2)\) with \(4\), we encounter \(8 \cdot 4\). Because multiplication takes precedence over addition, we solve this next.

Multiplying 8 by 4 results in 32, reducing the expression to \(3 + 32 + 11\). Remember, the hierarchy of operations ensures that you conduct these calculations before moving ahead with operations like addition or subtraction.

• Always perform multiplication (and division) from left to right if they appear sequentially in a problem.
• Failing to prioritize multiplication can lead to incorrect outcomes or misunderstandings in solving math problems.

Addition: The Final Step

Once you've dealt with both parentheses and multiplication, you can proceed to addition. Addition is typically the last operation you perform when solving a mathematical expression according to the order of operations.

In our exercise, after performing the necessary multiplication, the problem simplifies to \(3 + 32 + 11\). To complete this, we add from left to right as follows:

• First, add 3 and 32 to get 35.
• Then add 11 to 35 to arrive at the final result of 46.

Handling addition in this sequential manner allows us to maintain accuracy throughout our calculations. It is important to follow through with each step methodically, ensuring that each operation builds upon the one before it.