Problem 83 - Prealgebra

Question

Use the rule for order of operations to combine the following. \(3+2 \cdot 7\)

Step-by-Step Solution

Answer

The result is 17.

Step 1: Understand PEMDAS

To solve arithmetic expressions, we use the order of operations known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Identify any operations that need priority based on this rule.

Step 2: Identify Operations in the Expression

In the expression \(3 + 2 \cdot 7\), there are two operations: addition and multiplication. According to PEMDAS, multiplication takes priority over addition.

Step 3: Perform Multiplication First

Solve the multiplication part of the expression: \(2 \cdot 7 = 14\).

Step 4: Solve the Remaining Expression

After performing the multiplication, the expression simplifies to \(3 + 14\). Now, perform the addition.

Step 5: Addition

Add the numbers together: \(3 + 14 = 17\).

Key Concepts

Understanding the Order of Operations - PEMDAS

When tackling arithmetic expressions, it's crucial to use the right order of operations to achieve the correct answer. This rule is encapsulated in the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

PEMDAS helps us remember the sequence in which operations should be performed. Parentheses and exponents come first, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Grasping this concept ensures that mathematical expressions are simplified correctly, leading to accurate results.

Breaking Down Arithmetic Expressions

An arithmetic expression is a combination of numbers and operations such as addition, subtraction, multiplication, and division. These expressions must be solved carefully, respecting the rules of the order of operations. When you encounter an expression like \(3 + 2 \cdot 7\), it's important to pick apart each element to decide the order of calculation.

In this expression, we initially spot two operations: addition and multiplication. Each operation behaves differently in the context of PEMDAS, determining how and when they are performed. Knowing how to dissect expressions like these into their individual components is key to solving them accurately, avoiding common pitfalls often faced by students.

Prioritizing Multiplication in Expressions

In the hierarchy governed by PEMDAS, multiplication holds more weight than addition or subtraction. This means that when an expression contains both multiplication and addition, multiplication must be performed first.

Consider \(3 + 2 \cdot 7\). According to the rules, multiplication is executed first. Hence, you solve the \(2 \cdot 7\) part of the expression first, resulting in 14. Once all multiplication and division have been completed, only then do you proceed to the next level, which is addition and subtraction. This priority on multiplication ensures accuracy in calculations and helps to prevent mistakes.

Rules for Addition and Subtraction

Once any parentheses and priority operations like multiplication and division are handled, what's left in the expression are addition and subtraction. Both operations share the same level of precedence, which means they should be solved in the order they appear, from left to right.

For instance, after solving the multiplication in our expression \(3 + 2 \cdot 7 = 3 + 14\), we move on to simple addition: \(3 + 14\). If subtraction had been present in the expression, we would continue moving left to right, doing each operation in turn. Following this straightforward method maintains the natural flow of the arithmetic process, aiming towards an accurate final solution. This systematic approach clarifies the sometimes confusing operations of addition and subtraction.