Problem 24 - College Algebra

Question

Simplify the given expression. $$ (15-7) \cdot(3-7) $$

Step-by-Step Solution

Answer

The simplified expression is \(-32\).

Step 1: Evaluate the expression inside the first parenthesis

Start by simplifying the expression inside the first set of parentheses, which is \(15 - 7\). Calculate the difference to get \(8\).

Step 2: Evaluate the expression inside the second parenthesis

Next, move to the expression inside the second set of parentheses, which is \(3 - 7\). Calculate the difference to get \(-4\).

Step 3: Multiply the results from the parentheses

Now, take the simplified results from each parenthesis: \(8\) from the first and \(-4\) from the second. Multiply these two values together: \(8 \times (-4) = -32\).

Key Concepts

Order of Operations

When simplifying algebraic expressions, understanding the order of operations is crucial. This concept, often remembered by the acronym PEMDAS, guides us in which operations to perform first.
PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

First, always handle any operations inside parentheses. Only then do you move onto exponents, and so forth down the list.
In the exercise shared, the first thing we did was resolve the expressions within each set of parentheses. By doing so, we adhere to the order of operations, which ensures the correct result. This method prevents confusion and errors in problems that contain various operations.

Parentheses in Mathematics

Parentheses play a significant role in mathematics by prioritizing certain operations over others. When you see parentheses in an expression, it's a signal to perform the operations within them first. This rule helps maintain order and clarity in expressions that would otherwise be lengthy and potentially confusing.
In our exercise, parentheses helped group specific operations – specifically, subtraction in two distinct sets. We evaluated what's inside the parentheses to simplify the larger expression effectively. Ignoring or skipping this step could lead to incorrect results since operations within assume precedence over those outside.
Utilizing parentheses correctly can aid in organizing thoughts as you work through complex arithmetic and algebraic problems.

Arithmetic Operations

Arithmetic operations encompass the basic processes of addition, subtraction, multiplication, and division. Each operation serves a unique function, and understanding them is essential to solving algebraic expressions effectively.
In our exercise, we employed subtraction and multiplication:

• Subtraction allowed us to reduce numbers and simplify expressions within parentheses.
• Multiplication brought together the results of the previously simplified sections, leading to the final solution.

Having a firm grasp of these operations means recognizing when and how to use them, particularly as they relate to the order of operations. It's vital to treat these tasks sequentially, ensuring operations in parentheses are completed before moving elsewhere within the expression.