Problem 87 - Prealgebra

Question

Use the rule for order of operations to simplify each expression. $$22-2 \cdot 3$$

Step-by-Step Solution

Answer

The simplified expression is 16.

Step 1: Identify the Order of Operations

The order of operations can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means we need to perform Multiplication and Division before Addition and Subtraction.

Step 2: Multiplication

In the expression \(22 - 2 \cdot 3\), we first need to perform the multiplication. Calculate \(2 \cdot 3\) which equals 6.

Step 3: Subtraction

Substitute the product from Step 2 back into the expression: \(22 - 6\). Now perform the subtraction: \(22 - 6 = 16\).

Key Concepts

Understanding PEMDAS

When solving mathematical expressions, the order in which operations are performed is crucial to get the correct answer. PEMDAS is an acronym that helps us remember this order:

• Parentheses: Solve any operations inside parentheses first.
• Exponents: Next, handle exponents or powers.
• Multiplication and Division: These are performed from left to right.
• Addition and Subtraction: Again, these are performed from left to right.

This means Multiplication and Division have the same priority, as do Addition and Subtraction.
Following PEMDAS precisely is vital in simplifying expressions correctly.

The Role of Multiplication

Multiplication in the order of operations has a higher precedence than addition and subtraction.
In the expression you've been given, you must perform the multiplication before moving on to other operations.
In the example expression \(22 - 2 \cdot 3\), identify the multiplication process: \(2 \cdot 3\), which results in 6.
Remember that ignoring the multiplication or doing it out of order would lead to incorrect results.
Multiplication essentially combines groups of numbers. You can think of it as repeated addition, which helps visualize why it's done before additions in expressions.

Importance of Subtraction

Subtraction, like addition, is one of the basic arithmetic operations but is one of the last operations you complete when following the order defined by PEMDAS.
In your expression, after handling multiplication, you move to subtraction: \(22 - 6\). When subtracting, what you're doing is essentially finding the difference between two values.

• Subtracting can be viewed as adding a negative number.
• To simplify, consider how much is left after removing a certain quantity.

Correct subtraction is crucial because it results in the final answer of an operation.
Foregoing accuracy here would mean missing the whole essence of following the order and structured approach.

How to Simplify Expressions

Simplifying expressions effectively means reducing them to their simplest form or to a single numerical answer if possible.
Using PEMDAS is your map to navigate through the operations. For the expression \(22 - 2 \cdot 3\), by following the steps below, you simplify correctly:

• Identify and group different operations using PEMDAS.
• Always solve from left to right for operations with the same precedence like multiplication and division.
• For each step, simplify accordingly: perform operations and replace them within the expression.

The final aim is to reach an expression that's concise and correct, such as deriving \(22 - 6 = 16\).
Simplifying not only provides an answer but also helps in understanding the underlying math neatly.