Problem 17 - College Algebra

Question

For the following exercises, simplify the given expression. $$ 14 \cdot 3 \div 7-6 $$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Perform Multiplication

The given expression is \(14 \cdot 3 \div 7 - 6\). The first operation according to the order of operations (PEMDAS/BODMAS) is the multiplication \(14 \cdot 3\). This results in 42. Now, the expression is \(42 \div 7 - 6\).

Step 2: Perform Division

Next, we perform the division in the expression \(42 \div 7 - 6\). Dividing 42 by 7 results in 6. The expression now becomes \(6 - 6\).

Step 3: Perform Subtraction

Finally, perform the subtraction \(6 - 6\). Subtracting 6 from 6 results in 0. The expression simplifies to 0.

Key Concepts

Understanding the Order of Operations

When tackling algebraic expressions, especially those that involve multiple operations, it's crucial to follow the order of operations. This rule helps ensure consistency and accuracy in solving mathematical expressions. Peeking through the acronym PEMDAS/BODMAS can be quite helpful:

• P/B - Parentheses/Brackets: Complete any operations inside parentheses first.
• E/O - Exponents/Orders: Solve all exponents (powers and square roots, etc.) next.
• MD - Multiplication and Division: These are done from left to right as they appear in the expression.
• AS - Addition and Subtraction: Lastly, perform all additions and subtractions from left to right.

For our problem, we started with multiplication, followed by division, and then subtraction. This method helped ensure the correct simplification of the expression. By consistently using this sequence, you can simplify expressions accurately.

How to Simplify Algebraic Expressions

Simplification in algebra is about reducing expressions to their most basic form. The goal is to make them easier to understand and work with. For the expression given, simplification means:

• Carrying out operations step by step as dictated by the order of operations.
• Combining like terms and constants.
• Breaking an expression down to arrive at the simplest form.

In our example, after carrying out multiplication and division, the expression was reduced to a single subtraction operation. This process shows that by applying algebraic properties correctly, you transform complex expressions into simpler and more manageable forms.

Working with Arithmetic Expressions

Arithmetic expressions are combinations of numbers and operations such as addition, subtraction, multiplication, and division. Understanding how to handle these operations is fundamental in algebra.For example, the expression given: \(14 \cdot 3 \div 7 - 6\) involves basic arithmetic operations. When working with such expressions, you should:

• Identify and correctly apply each operation in sequence.
• Take note of any implied operations, such as multiplication in "14 \cdot 3".
• Acknowledge operations like subtraction as part of sequence completion.

Approaching arithmetic expressions with knowledge of both operations involved and their order allows for correct evaluations and interpretations.