Problem 76 - College Algebra

Question

State the name of the property illustrated. \(11 \cdot(7+4)-11 \cdot 7+11 \cdot 4\)

Step-by-Step Solution

Answer

The property illustrated is the distributive property.

Step 1: Identify the Property

The algebraic expression is \(11 \cdot(7+4)-11 \cdot 7+11 \cdot 4\). This expression can be rewritten as \(11 \cdot 11-11 \cdot 7+11 \cdot 4\). This is the form \(a(b+c) = ab + ac\), which is a form of the distributive property.

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

The property illustrated is the distributive property.

Key Concepts

Algebraic Properties

Algebraic properties are essential rules that form the backbone of algebra. They allow us to simplify expressions and solve equations efficiently. One of the most important properties is the Distributive Property, which combines multiplication and addition. This property states that if you have a number multiplying a sum, you can distribute the multiplication over each addend. For example, in the expression \( a(b+c) = ab + ac \), the distributive property is at work. It helps us distribute the multiplier \( a \) to both \( b \) and \( c \), simplifying the problem.
Understanding these properties helps in solving complex mathematical problems swiftly. They enable learners to recognize patterns and solutions in algebraic problems. The distributive property is particularly useful when dealing with large numbers or variables, as it breaks down tasks into smaller, simpler calculations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations. They differ from equations as they do not have an equality sign. Consider the expression \( 11 \cdot(7+4) \); it can be broken down and simplified using algebraic properties, like the distributive property, mentioned earlier.
Breaking down expressions into simpler parts is key to understanding and solving them. Each part of an expression, such as a term or factor, plays a crucial role in simplifying or evaluating it. Recognizing how to manipulate these expressions using algebraic principles transforms seemingly complex problems into manageable ones.

• Terms: Single numbers or variables, or the product of numbers and variables.
• Factors: Numbers or expressions being multiplied.
• Coefficients: Numbers paired with variables.

Comprehending these components offers a clearer view of how mathematical expressions work and how to handle them effectively.

College Algebra

College Algebra is an advanced study area that builds on fundamental arithmetic and algebraic concepts. Students encounter various algebraic properties, like the distributive property, honed at this level. This property is especially significant as it appears in numerous topics covered in college courses, such as polynomial multiplication, factoring, and algebraic expansions.
Conceptual mastery at this stage involves recognizing patterns and applying strategic rules to solve problems. College Algebra develops critical thinking by engaging students in complex problem-solving tasks, enabling them to apply algebraic rules thoroughly across various scenarios.
College Algebra provides a foundation for subjects like calculus and statistics, making an understanding of algebraic properties and mathematical expressions fundamental for further studies and applications in real-world problems. Students learn to see these principles not just as abstract rules but as tools to reach logical, functional conclusions in diverse fields of study.