Problem 71 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$2 z_{t}\left(L_{m}+8 k\right)$$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression is $$2 z_{t} L_{m} + 16 z_{t} k$$.

Step 1: Identify the terms to be distributed

We are given the expression $$2 z_{t}\left(L_{m}+8 k\right)$$. We will first identify the terms within the parentheses and the term outside the parentheses that will be distributed. The term outside the parentheses is $$2 z_{t}$$, and the terms inside the parentheses are $$L_{m}$$ and $$8 k$$.

Step 2: Apply the distributive property

We will now apply the distributive property by multiplying $$2 z_{t}$$ with each term inside the parentheses ($$L_{m}$$ and $$8 k$$) one by one.

Step 3: Multiply $$2 z_{t}$$ with $$L_{m}$$

Multiplying $$2 z_{t}$$ with $$L_{m}$$, we get: $$2 z_{t} * L_{m} = 2 z_{t} L_{m}$$.

Step 4: Multiply $$2 z_{t}$$ with $$8 k$$

Multiplying $$2 z_{t}$$ with $$8 k$$, we get: $$2 z_{t} * 8 k = 16 z_{t} k$$.

Step 5: Combine the products

Now, we combine the products from Steps 3 and 4 to get the final expanded expression: $$2 z_{t} L_{m} + 16 z_{t} k$$. So, the expression $$2 z_{t}\left(L_{m}+8 k\right)$$ when expanded using the distributive property is $$2 z_{t} L_{m} + 16 z_{t} k$$.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( z \), \( k \), or \( L_m \) in our exercise), and operators (such as addition, subtraction, multiplication, and division). The key feature of algebraic expressions is that they represent quantities in a generalized form, which makes them pivotal in algebra.

For instance, \( 2z_t(L_m + 8k) \) is an algebraic expression. Here, \( 2z_t \) is a coefficient that multiplies the variable \( z_t \) and \( L_m \) and \( 8k \) are terms within the parentheses. Understanding the structure of such expressions is crucial because it allows students to manipulate and simplify them using various algebraic principles, such as the distributive property, which we use to expand expressions.

Expanding Expressions

When it comes to expanding expressions, we often utilize the distributive property. This property stipulates that multiplying a sum by a number is the same as multiplying each addend by the number and then summing the products. Essentially, you distribute the multiplication over the addition within the parentheses.

In the given exercise, \( 2z_t(L_m + 8k) \) is expanded by distributing \( 2z_t \) across \( L_m \) and \( 8k \) resulting in \( 2z_tL_m + 16z_tk \). This process transforms the expression into a simpler, equivalent form that makes other operations such as addition, subtraction, or further simplification easier to handle. Developing proficiency in expanding expressions aids in solving algebraic equations and understanding functional relationships.

Elementary Algebra

Elementary algebra is the foundational branch of mathematics that deals with the use and manipulation of algebraic expressions and equations. It plays a crucial role in developing students' problem-solving abilities because it teaches them how to work with abstract concepts and represent real-world situations mathematically.

In our example, using the distributive property exemplifies an elementary algebra technique where algebraic expressions are manipulated to reveal a clearer structure or simplify calculations. As students progress in their mathematical education, understanding the rationale behind each step of such problem-solving strategies becomes key to mastering algebra and preparing for more advanced mathematical concepts.