Problem 11 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(16+2 \cdot(8-6)\)

Step-by-Step Solution

Answer

The value of the expression is 20.

Step 1: Evaluate the Expression Inside the Parentheses

Begin by calculating the expression inside the parentheses. In this problem, we have \(8 - 6\). Subtract 6 from 8 to get:\[8 - 6 = 2\]

Step 2: Apply the Order of Operations

Next, follow the order of operations (PEMDAS/BODMAS—Parentheses/Brackets, Exponents/Orders, Multiplication/Division, Addition/Subtraction). In the expression \(16 + 2 \cdot (8 - 6)\), after evaluating the parentheses, you are left with \(16 + 2 \cdot 2\). First, we perform the multiplication part:\[2 \cdot 2 = 4\]

Step 3: Perform the Addition

Finally, complete the calculation by adding the results of the previous operations. You have the expression \(16 + 4\). Simply add 4 to 16:\[16 + 4 = 20\]

Key Concepts

PEMDAS

PEMDAS is an acronym that helps students remember the order of operations in mathematical expressions. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following this specific order ensures that everyone solves expressions the same way, leading to the correct answer every time.

To use PEMDAS effectively:

• Start with any operations inside Parentheses.
• Then look for Exponents.
• Next, handle Multiplication and Division, moving from left to right across the expression.
• Finally, perform Addition and Subtraction, again moving from left to right.

Remember, PEMDAS is your guide through the maze of a math problem. It keeps your calculations structured and accurate.

Parentheses

Parentheses are not just brackets; they're VIPs in math expressions. Anything inside the Parentheses gets priority treatment. You start solving from the inside out.

In the expression given:
Given: \(8 - 6\) inside the parentheses:

• Calculate the value inside to simplify the expression further.
• Parentheses ensure you're clearly pointing out what needs to be solved first.

This highlights how operations in Parentheses are like a step inside a step.
Always handle them first to set the rest of the calculation process on the right path.

Multiplication

After dealing with the Parentheses, Multiplication comes into play. Following PEMDAS, you perform all Multiplications and Divisions next, once you've resolved the contents of any Parentheses.

Using the simplified expression: From step 1, after solving (8 - 6):

• You're left with \(16 + 2 \cdot 2\).
• The next step is to multiply \(2 \cdot 2\), which gives you 4.

Multiplication simplifies expressions by combining like terms through repeated addition. In this context, it takes the value found and drives it forward to be added to the next number.

Addition

Addition is often one of the final operations in a PEMDAS sequence, appearing after you've completed any Multiplications or Divisions.

In the expression: After multiplying, 16 + 4 remains:

• Addition links together separate parts of an expression to form a complete answer.
• By adding 16 and 4, you arrive at the final answer of 20.

Addition is straightforward and wraps up operations by culminating in the result. Though it comes last in this sequence, don't underestimate its power to bring all the previous calculations together.