Problem 17 - Precalculus Mathematics for Calculus

Question

Rewrite the expression using the given property of real numbers. Distributive Property, \(4(A+B)=\) _____

Step-by-Step Solution

Answer

\(4A + 4B\)

Step 1: Understand the Distributive Property

The distributive property allows you to distribute a multiplication operation across terms inside parentheses. For any real numbers \(a, b,\) and \(c\), it can be expressed as \(a(b + c) = ab + ac\). In this exercise, we will be distributing the number 4 across the terms inside the parentheses \((A + B)\).

Step 2: Apply the Distributive Property

Using the distributive property, multiply the number outside the parenthesis by each term inside the parenthesis. This means multiplying \(4\) by each of \(A\) and \(B\). So, \(4(A+B)\) becomes \(4 \times A + 4 \times B\).

Step 3: Simplify the Expression

Now, simplify the expression by performing the multiplication. This results in \(4A + 4B\). Thus, the expression \(4(A+B)\) rewritten using the distributive property is \(4A + 4B\).

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers you can think of that are not imaginary. This group of numbers encompasses:

• Natural numbers (like 1, 2, 3).
• Whole numbers (including 0 along with natural numbers).
• Integers (both positive and negative whole numbers).
• Rational numbers (fractions and numbers that can be expressed as a ratio of two integers).
• Irrational numbers (numbers that cannot be expressed as a simple fraction, like \(\pi\) and the square root of 2).

Real numbers can be found on a number line where both rational and irrational numbers exist. They are crucial in algebra and are used in everyday calculations.
Understanding real numbers helps in grasping how they interact in equations and expressions, like when using the distributive property. This sets the foundation for more complex mathematical concepts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They provide a way to generalize mathematical problems and relationships.
An expression like \(4(A + B)\) involves a number (4) and a sum (\(A + B\)), which are multiplied together. This is a typical algebraic expression with a real number coefficient (4) and variables (A and B).
To work with algebraic expressions, you need to understand the variables and be comfortable performing operations with them.
Knowing how to manipulate these expressions is key in solving algebraic equations, simplifying expressions, and expanding brackets using properties such as the distributive property.

• Simplification of expressions involves reducing them to their simplest form.
• When applying the distributive property to \(4(A + B)\), you distribute the multiplication over the terms within the parentheses.

This results in a simplified form, as shown in the solution: \(4A + 4B\).

Mathematical Properties

Mathematical properties are rules that define how numbers operate and relate to each other. They form the foundation for solving equations and manipulating expressions.
The distributive property is one of these essential properties, especially useful in algebra. It states that for any real numbers \(a, b,\) and \(c\), the product \(a(b + c)\) is equal to \(ab + ac\). This property is used to simplify and solve expressions.

• It helps in eliminating parentheses by distributing a factor across terms within them.
• This property maintains the equality of expressions while breaking down complex expressions into simpler ones.

By applying the distributive property in the exercise \(4(A + B)\), each term inside the parentheses gets multiplied by 4, resulting in \(4A + 4B\).
Understanding and applying such properties correctly is crucial in mastering algebra and advancing to higher-level mathematics.