Problem 58 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$(1+d) e$$

Step-by-Step Solution

Answer

Question: Expand the expression \((1 + d)e\) using the distributive property. Answer: \(e + ed\)

Step 1: Identify the terms

Let's identify the terms in the expression \((1+d)e\). Here, the real number \(e\) is being multiplied by the quantity inside the parentheses, which consists of \(1\) and \(d\). So, we have: - \(a = e\) - \(b = 1\) - \(c = d\) Now, we need to apply the distributive property to these terms.

Step 2: Apply the distributive property

According to the distributive property, \(a(b+c) = ab + ac\). So, we will multiply \(e\) by each term inside the parentheses and add the results: $$e(1+d) = e\cdot 1 + e \cdot d$$

Step 3: Simplify the expression

Now, we will simplify the expression. Since \(e \cdot 1 = e\), the simplified expression is: $$e(1+d) = e + ed$$ The expanded quantity using the distributive property is \(e + ed\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial in the study of algebra. These expressions are combinations of numbers, variables (like \(d\) and \(e\) in our exercise), and arithmetic operations such as addition, subtraction, multiplication, and division. In the context of our exercise \((1+d)e\), the expression consists of a variable \(d\), a constant \(1\), and a multiplication operation involving the variable \(e\). The key to working with algebraic expressions is recognizing the parts and knowing the rules that govern how these parts can interact or be transformed.

For instance, when we multiply \(e\) by the quantity inside the parentheses, we're combining the constant and variable to form a new expression. Keeping track of each part ensures we apply the correct algebraic principles, like the distributive property, to expand and simplify the expressions accurately.

Expanding Expressions

Expanding expressions involves rewriting an algebraic expression in an equivalent form where all possible multiplication has been carried out. The distributive property is often used in this process. It allows us to remove parentheses by distributing a single term over a sum or difference inside the parentheses, as seen in the step-by-step solution for the expression \((1+d)e\).

By doing so, it becomes easier to combine like terms or further manipulate the expression for solving equations or inequalities. Expanding expressions is a fundamental skill in algebra, as it sets the stage for understanding more complex operations and solving various types of algebraic problems.

Simplifying Expressions

Once an expression has been expanded, the next step in the process is typically to simplify it. Simplifying an algebraic expression means to rewrite it in its most basic or compact form, combining like terms and reducing coefficients as much as possible. In our exercise, after expanding \((1+d)e\) to \(e + ed\), no further simplification is necessary because there are no like terms to combine or coefficients to reduce. However, in more complex expressions, simplification could involve combining several terms, factoring, or even canceling terms to achieve the simplest form possible.

To simplify effectively, knowledge of the order of operations is vital, as it guides us on which steps to take first when reducing an expression.

Elementary Algebra

Elementary algebra encompasses the basic concepts of algebra, including understanding and manipulating algebraic expressions using foundational properties such as the distributive property. It's these principles that form the basis of algebra and allow us to solve equations, work with inequalities, and analyze functions.

Through exercises like the one presented, \((1+d)e\), students learn to apply these rules systematically, developing an intuition for how algebraic expressions can be transformed and simplified. This foundational knowledge is imperative as it paves the way for tackling more sophisticated mathematical concepts and problems in the future.