Problem 76 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the expressions. $$ 10 a_{z}\left(b_{z}+c\right) $$

Step-by-Step Solution

Answer

Question: Expand the expression using the distributive property: \(10a_{z}(b_{z}+c)\) Answer: \(10a_{z}b_{z} + 10a_{z}c\)

Step 1: Identify the terms to be distributed

In the expression \(10a_{z}(b_{z}+c)\), the term \(10a_{z}\) will be distributed across the sum \((b_{z}+c)\).

Step 2: Apply the distributive property

Use the distributive property by multiplying \(10a_{z}\) by both \(b_{z}\) and \(c\) separately. So, \(10a_{z}(b_{z}+c) = 10a_{z} \cdot b_{z} + 10a_{z} \cdot c\).

Step 3: Write the expanded expression

The expanded expression after applying the distributive property is: \(10a_{z}b_{z} + 10a_{z}c\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical statement composed of numbers, variables, and arithmetic operators like addition, subtraction, multiplication, and division. They represent values in an abstract form. For example, the expression \(10a_{z}(b_{z}+c)\) is an algebraic expression. It includes:

• Coefficients: Numbers in front of variables (e.g., \(10\) in \(10a_{z}\))
• Variables: Letters representing unknown values (e.g., \(a_{z}\) and \(b_{z}\))
• Arithmetic operations: Indicated by symbols like \(+\), \(-\), or \(\times\)

Understanding algebraic expressions is crucial because they form the basis of more complex algebraic equations and problems. They often require simplification or expansion, which involves rewriting them in a form that's easier to interpret or solve.

Expansion

Expansion in algebra refers to the process of simplifying an expression to remove parentheses by using the distributive property or other algebraic properties. This process converts expressions from a compact form to a more expanded view. Let's take the example from our exercise:
When we have the expression \(10a_{z}(b_{z} + c)\), the goal of expansion is to simplify it by distributing \(10a_{z}\) to both terms inside the parentheses:

• Multiply \(10a_{z}\) by \(b_{z}\), resulting in \(10a_{z}b_{z}\)
• Multiply \(10a_{z}\) by \(c\), resulting in \(10a_{z}c\)

Therefore, after expanding, we get the expression \(10a_{z}b_{z} + 10a_{z}c\). This process helps in dealing with expressions during algebraic operations, making computations straightforward and solutions more apparent.

Mathematical Distributive Law

The distributive property is a fundamental law in algebra that helps us simplify complex expressions and equations. It states that a term multiplied by the sum of two addends can be expressed as the sum of the term multiplied by each addend individually. In formulaic terms, this can be written as:
\[ a(b + c) = ab + ac \]
Applying this law simplifies expressions like \(10a_{z}(b_{z}+c)\) by distributing the term \(10a_{z}\) to both \(b_{z}\) and \(c\), which results in \(10a_{z}b_{z} + 10a_{z}c\). The advantages of using the distributive property include:

• Reduces the number of terms or components in a problem, making it less complex
• Essential for solving equations or simplifying expressions in algebra
• Facilitates mental math by breaking down large numbers into smaller, more manageable pieces

Understanding and applying the distributive law is critical for success in algebra, as it underpins many operations and equations encountered in mathematical problem-solving.