Problem 40 - Elementary and Intermediate Algebra

Question

Simplify. $$ (8-2 \cdot 3)-9 $$

Step-by-Step Solution

Answer

-7

Step 1: Evaluate the Multiplication

First, handle the multiplication within the parentheses. Compute the value of \(2 \cdot 3\).

Step 2: Compute the product

Calculate \(2 \cdot 3 = 6\).

Step 2: Substitute the Result

Replace \(2 \cdot 3\) with 6 in the original expression, which gives: \( (8 - 6) - 9 \).

Step 3: Evaluate Inside Parentheses

Next, solve the expression inside the parentheses. Compute \(8 - 6\).

Step 5: Calculate the Difference

Calculate \(8 - 6 = 2\).

Step 4: Perform the Final Subtraction

Finally, subtract 9 from the result. Compute \(2 - 9\).

Step 7: Compute the Final Result

Calculate \(2 - 9 = -7\).

Key Concepts

Simplification

Simplification is an essential process in mathematics, helping you to reduce complex expressions into simpler forms. In the given exercise, simplification is achieved through a series of operations, following precise rules known as the order of operations, or BIDMAS/BODMAS (Brackets, Orders (i.e. powers and roots, etc.), Division and Multiplication, Addition and Subtraction). The goal is to make solving the problem more manageable.
Always simplify expressions step-by-step, starting with the operations inside the parentheses. Then, proceed by performing multiplication and division from left to right, followed by addition and subtraction from left to right.
Without a proper understanding of simplification, math problems can become cumbersome and more error-prone.

Multiplication

Multiplication is one of the fundamental arithmetic operations, essential for simplifying expressions. In our exercise, multiplication appears as part of the term inside the parentheses: \( 2 \cdot 3 \). According to the order of operations, multiplication comes after parentheses but before addition and subtraction.
Here, to simplify \( (8-2 \cdot 3)-9 \), you first calculate the product \( 2 \cdot 3 \). This multiplication converts the term into a simpler number, 6.
Without correctly performing this multiplication first, the resulting simplification would be inaccurate. Always ensure to handle multiplication operations immediately after dealing with any parentheses.

Subtraction

Subtraction follows multiplication and division within the order of operations. In our exercise, once we've handled the multiplication inside the parentheses and simplified it to \( 8-6 \), the next step is performing the subtraction.
Compute \( 8 - 6 \) to obtain 2. Finally, subtract 9 from the result: \( 2 - 9 \).
Subtraction can seem straightforward, but it's crucial to perform it in the correct sequence—after any multiplications and when handling expressions within the parentheses. Missing this step can lead to incorrect results, making it vital to pay close attention to subtraction in complex expressions.

Parentheses

Parentheses indicate that the operations within them should be performed first. In mathematical expressions like \( (8-2 \cdot 3)-9 \), it's crucial to resolve everything within the parentheses before tackling outside operations.
Begin by handling any operations inside the parentheses, like multiplication: \( 2 \cdot 3 = 6 \). This changes the expression to \( (8-6)-9 \).
After resolving the initial subtraction within the parentheses, the expression becomes easier to manage: \( 2-9 \). Treating parentheses correctly ensures you simplify expressions correctly and avoid mistakes.
Always look out for parentheses and remember to simplify what's inside them first, following the order of operations.