Problem 26 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (w+6) 4 $$

Step-by-Step Solution

Answer

The expression without parentheses is 4w + 24.

Step 1: Applying Distributive Property

To apply the distributive property, multiply each term inside of the parentheses by 4. The distributive property can be written as: a(b + c) = ab + ac. So, applying this to our problem we get: 4*(w+6) = 4w + 4*6

Step 2: Evaluating Expression

Now, solve the multiplication operation 4*6. This gives us 4w + 24.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. It represents a mathematical phrase that can involve addition, subtraction, multiplication, or division. In the expression \((w+6)\), \(w\) is a variable, and 6 is a constant. The way they are placed inside the parentheses indicates the operations that need to be performed.Variables are symbols, often letters, that represent unknown or varying numbers. They allow us to generalize mathematical principles by replacing specific values with placeholders, which can be any number.

• Translation Example: In an algebraic expression like \(w+6\), \(w\) can take different values, depending on the problem context.
• Importance: Understanding how variables and constants work together helps simplify problems and predictions.

Multiplication

Multiplication is one of the basic operations in mathematics, symbolized by \(*\) or sometimes left implied by placing two terms next to each other. When applying multiplication within algebraic expressions, each term inside parentheses is multiplied by the outside term.In our example, we multiply each element within \((w+6)\) by 4:

• Multiply \(w\) by 4 to get \(4w\).
• Multiply 6 by 4 to get 24.

This operation distributes the multiplication across the terms, ensuring each component is equally affected. This approach is useful because it allows for simplifying expressions and solving equations more easily.Tips for mastering multiplication with algebra:- Always remember to multiply each term inside the parentheses by the factor outside.- Practice by applying it to different expressions to gain confidence.

Mathematical Properties

Mathematical properties are rules that hold for various operations, ensuring consistency and predictability in calculations. One of the fundamental properties used in algebra is the Distributive Property.The Distributive Property shows how multiplication interacts with addition inside parentheses. Its general form is \(a(b + c) = ab + ac\). This property allows us to eliminate parentheses by distributing the multiplicative factor over each addition term inside.

• Example Usage: For \((w+6)4\), applying the Distributive Property gives us \(4w + 24\).
• Why It Matters: It simplifies complex expressions and aids in solving equations.

Understanding mathematical properties like the Distributive Property provides a toolkit for manipulating and breaking down complex expressions, making Algebra a much more manageable subject. Also, it reinforces the idea that rules in mathematics have universal applications, leading to broader problem-solving skills.Learning these properties not only simplifies mathematics but also enhances logical reasoning.