Problem 61 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 13+(7-2)(5-1) $$

Step-by-Step Solution

Answer

The simplified expression is 33.

Step 1: Solve the Inner Expression

Identify the expression inside the parentheses and solve it first. There are two expressions: \((7-2)\) and \((5-1)\). Start with these calculations:\[7-2=5\]\[5-1=4\]

Step 2: Multiply the Results

Next, multiply the results of the two expressions solved in Step 1. The expression becomes:\[(7-2)(5-1) = 5 \times 4 = 20\]

Step 3: Add to the Initial Number

Now, add this result to the initial number in the expression. The full expression now simplifies to:\[13 + 20 = 33\]

Step 4: Final Solution

The simplified expression is:\[33\]

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying numerical expressions. The order of operations is a set of rules that state the best approach to solve an arithmetic expression. It helps you know which operation to perform first so that you achieve the correct result. The acronym PEMDAS is often used to remember the order:

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

In the provided exercise, the order of operations guides us to solve what is inside the parentheses first before moving to multiplication and then addition. Following this order ensures that everyone reaches the same correct solution. Always keep PEMDAS in mind when faced with a complex expression!

Parentheses

Parentheses are vital in numerical expressions as they indicate which operations should be completed first. They essentially alter the natural order of arithmetic operations to follow specific sequences. This prioritization inside parentheses ensures clarity and avoids any misinterpretation of an arithmetic expression.For example, in the expression given, \[13+(7-2)(5-1)\]we must first solve \[(7-2)\] and \[(5-1)\] before engaging with other operations. Without resolving these expressions in parentheses first, we might accidentally solve the expression wrongly. Utilizing parentheses correctly ensures accurate and orderly problem-solving. This illustrates the power of parentheses to focus on specific parts of an expression first.

Arithmetic Operations

Arithmetic operations form the backbone of simplifying expressions. They encompass basic mathematical operations like addition, subtraction, multiplication, and division. These operations are the building blocks of math, and understanding them is essential for solving any numerical expression.In our example expression, we have several arithmetic operations at play:

• Subtraction in the parentheses: \[(7-2) \] and \[(5-1)\]
• Multiplication of the results: \[5 \times 4\]
• Addition of the multiplication result to the initial number: \[13 + 20\]

Each of these operations plays a crucial role in the simplification process, and handling them step by step ensures a correct outcome. By mastering these basic operations, simplifying complex expressions becomes manageable and less daunting.