Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$8(2 x+3)$$

Step-by-Step Solution

Answer

The expression \( 8(2x+3) \) can be rewritten without parentheses as \( 16x + 24 \).

Step 1: Identify the expression inside the parentheses

The expression inside the parentheses is \( 2x + 3 \).

Step 2: Apply the distributive property

The distributive property states that for any real numbers a, b and c, the expression \( a(b + c) \) equals to \( ab + ac \). Therefore, distribute the 8 to every term inside the parentheses to get \( 8 * 2x + 8 * 3 \).

Step 3: Simplify the resulting expression

By doing the multiplication, we get \(16x + 24 \). This is the simplified form of the provided expression \(8(2 x+3)\).

Key Concepts

Simplify Algebraic Expressions

Understanding how to simplify algebraic expressions is a foundational element in algebra. When we simplify an expression, we aim to make it as straightforward as possible, often to make calculations easier or to identify like terms. Simplifying can involve several steps, from performing arithmetic operations to combining like terms.

For example, the expression \(8(2x+3)\) may look complicated with its parentheses, but we can make it simpler. When simplifying the expression, we have to ensure that we follow the order of operations and apply any relevant algebraic properties, such as the distributive property, to break down the expression into a more manageable form.

To practice this skill, you could try simplifying \(5(x+7) - 2(x-4)\) as an exercise. You would first distribute the 5 and -2 to the terms within their respective parentheses and then combine like terms.

Applying Distributive Property

The distributive property is a critical concept in algebra that allows us to multiply a single term by each term within a set of parentheses. In the context of the exercise \(8(2x+3)\), the distributive property is used to remove the parentheses and simplify the expression.

To apply the distributive property effectively, we follow this principle: for any real numbers a, b, and c, the property tells us that \(a(b + c) = ab + ac\). It is essentially about ensuring that the term outside the parentheses multiplies by each term inside, which extends to algebraic expressions. So, for our example, we distribute the 8 to both the 2x and the 3, resulting in \(16x + 24\) after simplification.

This process is used in various algebraic situations, such as expanding expressions or solving equations. To strengthen your understanding, you can practice by distributing the terms in an expression like \(3(x - 5) + 4(2x + 1)\) and then simplifying the result.

Basic Algebra Skills

Building basic algebra skills provides a sturdy foundation for all future math learning. These skills include understanding variables and constants, recognizing the various properties of numbers (like the distributive, associative, and commutative properties), and performing operations with algebraic expressions efficiently.

For students to excel in algebra, it’s essential to be comfortable with simplifying expressions, solving equations, and factoring. Regular practice with problems of varying difficulty, such as \(4x - 3(2x - 7) + 5\) or even \(x(5x^2 - 3x + 2)\) is crucial. Moreover, learning to visualize the problem and progressively working towards the solution, like identifying the distributive property in the given exercise \(8(2x+3)\) and simplifying it with methodical steps, is the approach to developing strong algebra skills.