Problem 66 - Prealgebra

Question

Apply the distributive property to each of the following expressions. $$4(2 a-5)$$

Step-by-Step Solution

Answer

The expression simplifies to \(8a - 20\).

Step 1: Identify the Expression

The given expression to apply the distributive property is \(4(2a - 5)\). This is in the form of \(a(b+c)\), where \(a = 4\), \(b = 2a\), and \(c = -5\).

Step 2: Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). Apply this to the expression: \(4(2a) + 4(-5)\).

Step 3: Simplify Each Term

Calculate \(4 \times 2a\) which is \(8a\), and \(4 \times (-5)\) which is \(-20\).

Step 4: Write the Final Expression

Combine the simplified terms: \(8a - 20\).

Key Concepts

Algebraic Expressions

In mathematics, understanding algebraic expressions is a cornerstone that helps in solving equations and performing calculations. An algebraic expression can contain numbers, variables, and arithmetic operations. These expressions are like sentences that tell you how numbers are related to one another.

For example, in the expression \(4(2a - 5)\), we have an expression inside parentheses \((2a - 5)\), multiplied by 4. In such expressions, the numbers (4, 2, and -5) are called coefficients and constants, while the letter \(a\) represents a variable, which can take on different values.

Algebraic expressions are essential because they allow us to generalize arithmetic, solve problems with unknowns, and form the basis of equations. Understanding how to manipulate these expressions is an important skill in algebra.

Prealgebra

Prealgebra serves as the foundation for all higher-level math, where key concepts such as the distributive property, variables, and basic operations are introduced. This stage is crucial for students transitioning from arithmetic to algebra by building necessary skills.

The distributive property is often introduced at this stage. It states that given an expression of the form \(a(b+c)\), you multiply \(a\) by both \(b\) and \(c\) separately. This concept is fundamental in prealgebra because it allows students to understand how to break down complex problems into simpler parts.

By practicing prealgebra, students develop an understanding of how numbers and operations interact, preparing them for algebra, where solving equations and manipulating expressions becomes more complex. This prepares students to approach problems like \(4(2a - 5)\) with confidence through systematic methods.

Simplifying Expressions

Simplifying expressions means transforming them into their simplest form without changing their value. It makes complex expressions more manageable and easier to understand or solve. To simplify, we often use properties such as the distributive property, combine like terms, and reduce fractions if necessary.

Take the expression \(4(2a - 5)\); first, we use the distributive property: multiply 4 with each term inside the parentheses. This gives us \(8a - 20\).

Simplifying doesn’t just end here. If there are like terms and similar factors, we combine them for clarity and compactness. For expressions involving variables, simplification might also involve factoring or canceling out terms. These steps ensure the expression is as straightforward as it can be, which is particularly beneficial in solving equations or comparing expressions.

Through practice, mastering simplification helps streamline problem-solving in algebra and beyond.