Problem 65 - Algebra 1

Question

Apply the distributive property. $$-5 w(-3+2 w)$$

Step-by-Step Solution

Answer

The simplified expression after applying the distributive property to \(−5w(−3 +2w)\) is \(15w -10w^2\).

Step 1: Identify terms

Identify the terms in the parentheses to which the distributive property will be applied, which are -3 and 2w.

Step 2: Apply distributive property

The distributive property states that the product of a number and a sum, \(a(b + c)\), is equal to the sum of the products of the number and each term in the sum, \(ab + ac\). Apply the distributive property to \(−5w(−3 +2w)\), which results in \(-5w*-3 + -5w*2w\).

Step 3: Simplify the expression

Perform the multiplication operations to simplify the expression to \(15w -10w^2\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is an essential skill that helps students manage complex algebraic expressions more easily. The process involves rewriting an expression in a form that is easier to work with—and often more compact—while keeping the original value unchanged.
Often in algebra, we encounter expressions that contain multiple terms connected by addition, subtraction, multiplication, or division. In the case of the problem, the expression

• \(-5w(-3+2w)\) consists of terms within a parenthesis that need to be simplified. Simplifying such expressions often involves the distributive property, which we will discuss further here.
  It is also important to combine like terms and carry out basic arithmetic operations to reduce expressions to their simplest form. This step makes it easier to evaluate algebraic expressions and solve equations. In the simplified form of
  • \(-5w \cdot -3 + -5w \cdot 2w\),
  we perform multiplication and then add the results to get a simplified result like \(15w - 10w^2\).

Understanding simplification will enhance your ability to manipulate algebraic expressions confidently.

Algebraic Expressions

Algebraic expressions are a foundational element of algebra. They consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, or division. An algebraic expression can be as simple as a single number or variable, such as

• \(5w\),

or as complex as

• \(-5w(-3+2w)\).

Algebraic expressions do not have an equal sign, which means they represent a value but not an equation. In the example

• \(-5w(-3+2w)\),

the expression represents a series of calculations that need to be done in sequence using properties like the distributive property.
To work effectively with algebraic expressions, one must understand how to handle both constants like \(-3\) and terms with variables like \(2w\). When simplifying, variables are combined based on specific rules, like combining similar terms or multiplying a term with multiple occurrences of the variable. This understanding lays the foundation for solving equations and inequalities.

Multiplication

Multiplication is one of the core operations in algebra, especially when simplifying expressions. It involves repeated addition of a number, usually known as the multiplier, a certain number of times. In the context of algebra, multiplication also extends to expressions that involve variables.
When dealing with expressions such as

• \(-5w(-3+2w)\),

we use multiplication to apply the distributive property across the terms within the parenthesis. Here,

• \(-5w\)

needs to be multiplied by each term inside the parenthesis \((-3\) and \(2w\)).
Here is how the multiplication happens:

• First, \(-5w \cdot -3 = 15w\) because multiplying two negative numbers results in a positive number.
• Second, \(-5w \cdot 2w = -10w^2\) where, for multiplication of variables, we add the exponents (here \(w\) becomes \(w^2\)).

This step-by-step multiplication is critical in using operations effectively within algebraic expressions, leading to simplified expressions and clearer results.