Problem 19 - College Algebra

Question

Simplify the given expression. $$ 6+2 \cdot 2-1 $$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Order of Operations

According to the order of operations (PEMDAS/BODMAS), multiplication and division should be done before addition and subtraction. So, we first identify any multiplication in the expression.

Step 2: Perform Multiplication

Identify the multiplication operation: \(2 \cdot 2\). Calculate this multiplication first.\[2 \cdot 2 = 4\]

Step 3: Substitute Back

Substitute the result of the multiplication back into the expression. Now the expression looks like this: \(6 + 4 - 1\).

Step 4: Perform Addition

Perform the addition operation next. Add \(6\) and \(4\) together.\[6 + 4 = 10\]

Step 5: Perform Subtraction

Finally, perform the subtraction. Subtract \(1\) from \(10\).\[10 - 1 = 9\]

Key Concepts

PEMDAS/BODMAS

When tackling arithmetic expressions, one crucial concept to understand is the order in which mathematical operations should be performed. This order is governed by the rules of PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In some countries, BODMAS is used instead, where 'B' stands for Brackets and 'O' for Orders (which is another term for exponents).

These acronyms help students remember the hierarchy of operations within a mathematical expression. It ensures expressions are simplified correctly.

For instance, in the expression given, the multiplication must be performed before addition and subtraction due to PEMDAS. Recognizing these operations in the correct sequence prevents mistakes and leads to an accurate solution.

Arithmetic Expressions

An arithmetic expression consists of numbers and operations such as addition, subtraction, multiplication, and division. These expressions can be simple like the one in our example or involve more complex operations, including parentheses and exponents. Understanding how to evaluate these expressions is an important foundational skill in mathematics.

In the exercise, the expression \(6 + 2 \cdot 2 - 1\) includes mixed operations. This particular expression does not contain parentheses or exponents, making it simpler. However, applying the correct order of operations is still crucial for arriving at the correct answer. With the basics of arithmetic expressions in mind, students can systematically work through each step using the order of operations.

Simplification Steps

Breaking down an expression into manageable steps makes simplifying it easier. Here's how the process can unfold:

• First, identify operations that need to be done, starting with multiplication and division if present.
• Perform the identified operation—in our case, that meant multiplying \(2 \cdot 2\), which results in \(4\).
• Substitute the result back into the expression, leaving \(6 + 4 - 1\).
• Proceed to addition: add \(6\) and \(4\) to get \(10\).
• Finally, carry out subtraction: subtract \(1\) from \(10\), yielding the final result of \(9\).

By following these simplification steps carefully and in the correct order, students can avoid mistakes and confidently solve similar problems. Each step builds upon the last, highlighting the importance of understanding and applying the order of operations correctly.