Problem 55 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Expand the expression using the distributive property: \(k(j + 1)\) Answer: The expanded form of the expression is \(kj + k\).

Step 1: Identify the terms

In this expression, k is the term outside the parenthesis and (j + 1) is inside the parenthesis.

Step 2: Apply the distributive property

Now we need to distribute k to both j and 1 inside the parenthesis. According to the distributive property, we have: $$k(j + 1) = k \cdot j + k \cdot 1$$

Step 3: Simplify the expression

We can now simplify the expression by performing the multiplication: $$k \cdot j + k \cdot 1 = kj + k$$ So, the expanded form of the given expression using the distributive property is: $$k(j + 1) = kj + k$$

Key Concepts

Elementary Algebra

Elementary algebra is the foundational stone of mathematics that deals with the manipulation of algebraic expressions—symbols and numbers combined using mathematical operations. This branch of mathematics introduces variables representing numbers, enabling the formulation of algebraic equations that can be solved to find unknown values.

It's essential for students to learn how to work with different forms of algebraic statements, particularly in performing operations such as addition, subtraction, multiplication, and division with variables. The distributive property, a critical concept in algebra, allows for the multiplying of a single term by each term within a parenthesis, thus 'distributing' the multiplication across the terms inside.

Expanding Expressions

Expanding expressions is a technique used to rewrite expressions by eliminating parentheses. This process is often done using the distributive property. In expanding, we take each term inside the parentheses, multiply it by the term outside, and then add the products.

To fully grasp this, imagine breaking down a compact package into individual elements to see all components clearly. For example, in the expression \(k(j+1)\), you are tasked to multiply \(k\) by each term inside the parentheses—this act of 'expansion' ultimately simplifies the expression and is a frequent operation in algebra.

Simplifying Expressions

Simplifying expressions involves reducing algebraic expressions to their simplest form. This typically includes combining like terms, using the distributive property to remove parentheses, and cancelling out terms when possible. The goal is to make the expression as compact and as easy to understand as possible without changing its value.

Let's consider our example \(k(j+1)\). After applying the distributive property, we get \(kj + k\). This is a simplified version because there's no further reduction or combination of like terms possible. Simplification makes algebraic expressions more manageable and prepares them for solving—if solving is required.

Multiplication

Multiplication is one of the four elementary mathematical operations and serves as a shortcut for repeated addition. In algebra, multiplication involving variables extends beyond mere numbers. Variables represent quantities that can vary, and when we multiply variables, we follow specific rules such as the distributive property.

In our example, \(k\) being multiplied by \( j+1\) demonstrates these rules at work. We calculate \(k\) times \(j\) and \(k\) times \(1\), then add the results. This process underpins many areas of algebra and is a skill that's essential for solving more complex mathematical problems.