Problem 25 - Intermediate Algebra

Question

Complete each statement so that the indicated property is illustrated. See Example 1. $3(2+d)= _____ Distributive property

Step-by-Step Solution

Answer

3(2 + d) = 6 + 3d, using the Distributive Property.

Step 1: Identify the components

The expression given is \(3(2 + d)\). In this expression, the number 3 is being multiplied by each term inside the parentheses (2 and \(d\)). This sets us up for using the Distributive Property.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). By applying this to \(3(2 + d)\), we multiply both 2 and \(d\) by 3. This gives us \(3 \cdot 2 + 3 \cdot d\).

Step 3: Simplify the expression

Calculate \(3 \times 2\) to get 6. Then, express \(3 \cdot d\) as \(3d\). This simplifies the distributed expression to \(6 + 3d\).

Key Concepts

Algebra

Algebra is like a magical tool in mathematics that helps us understand and work with numbers and variables. Variables are symbols, often letters like \(d\), that represent unknown values. Algebra allows us to create equations and expressions to solve problems more easily.
In the exercise given, we have an algebraic expression: \(3(2 + d)\). This is an example of an algebraic expression because it includes numbers, variables, and operations (addition and multiplication) working together to describe a mathematical situation.
One big part of algebra is learning how to manipulate these expressions to help us find solutions or find equivalent expressions. In our case, this involves using properties, like the Distributive Property, to simplify or expand expressions. Understanding these properties and how they help reshape expressions is key to mastering algebra.

Simplification

Simplification is the process of transforming a complex expression into a simpler and more usable form. This is especially helpful in algebra, as it allows us to work with more manageable equations or expressions.
For the example \(3(2 + d)\), simplification means breaking down the expression into a form that’s easier to understand at a glance. Initially, it looks a bit tangled because of the parentheses and multiple terms. But by using the Distributive Property, we simplify it to \(6 + 3d\).
Let's break down why this is important:

• Creates clearer and more concise expressions.
• Makes it easier to solve equations or understand relationships between variables and constants.
• Helps identify and eliminate unnecessary components—making calculations quicker.

Simplifying might involve distributing terms, like we did here, or combining like terms to consolidate parts of an equation. This skill is foundational not only in algebra but in all of mathematics as it keeps expressions neat and functional.

Multiplication Properties

Multiplication properties are rules that help us handle multiplication in various mathematical situations, ensuring we can work accurately and efficiently with numbers and variables.
In the context of the exercise, we see the Distributive Property in action. This is a crucial property of multiplication, and it tells us that when you multiply a sum by a number, it's the same as multiplying each addend by the number and then adding the products. In simpler terms, the property allows you to "distribute" the multiplication over the addition inside the parentheses:

• The general formula is \(a(b + c) = ab + ac\).
• In our example, \(3(2 + d)\) becomes \(3 \cdot 2 + 3 \cdot d\), resulting in the expression \(6 + 3d\).
• It helps in simplifying expressions and solving equations by breaking down multiplication into more manageable parts.

These properties form the backbone of many algebraic processes, helping to keep our work consistent and reliable as we move through more complicated mathematical tasks. Remember, mastering these properties isn't just about memorizing formulas—it's about understanding how they transform expressions and improve our ability to solve problems.