Problem 17 - Algebra and Trigonometry

Question

\(15-20\) : Use properties of real numbers to write the expression without parentheses. $$ 4(2 m) $$

Step-by-Step Solution

Answer

The expression simplifies to \(8m\).

Step 1: Understand the Exercise

We need to simplify the expression \(4(2m)\) using the properties of real numbers to eliminate the parentheses.

Step 2: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). In this case, since there's only one term \(2m\) inside the parentheses, we distribute by multiplying \(4\) by \(2m\).

Step 3: Simplify the Expression

Multiply \(4\) by \(2m\). Calculation:\[ 4 \times 2m = 8m \]

Step 4: Write the Final Expression

The expression \(4(2m)\) simplifies to \(8m\) without any parentheses.

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra. It allows us to remove parentheses by distributing, or multiplying a factor outside the parentheses across each term within them. For the expression \(4(2m)\), the distributive property simplifies the process of multiplication. Even though the expression inside the parentheses is just one term, the distributive property is applied as follows:

• Multiply the number outside the parentheses, \(4\), by each term inside, which in this case is \(2m\).
• This results in \(4 \times 2m = 8m\).

This property is vital in algebra as it helps in simplifying expressions and solving equations with multiple terms.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves rewriting expressions in a more straightforward form. The goal is to make expressions easier to understand or solve by combining like terms and applying mathematical properties. When given an expression such as \(4(2m)\), simplifying involves three main steps:

• Use the distributive property to remove any parentheses, making the terms more navigable.
• Perform the multiplication to deal with each term independently. In this case, \(4 \times 2m = 8m\).
• Write the simplified version without unnecessary components like parentheses, which in this example results in simply \(8m\).

This method of simplification is common, allowing complex equations to be broken down into more manageable parts.

Properties of Real Numbers

The properties of real numbers form the foundation of many algebraic operations. These properties include the commutative, associative, and distributive properties. Here, we focus on how these help in the multiplication of numbers and variables:

• Commutative Property: This states that numbers can be multiplied in any order, e.g., \(a \times b = b \times a\). However, it wasn't directly used in simplifying \(4(2m)\).
• Associative Property: This deals with grouping numbers, where \((a \times b) \times c = a \times (b \times c)\). Again, this isn't directly applied here but underpins many mathematical operations.
• Distributive Property: We've already discussed this, but it basically means that \(a(b+c) = ab + ac\), which is pivotal in simplifying \(4(2m)\) to \(8m\).

Understanding these properties allows for greater flexibility and capability in algebraic manipulations, ensuring that expressions and equations are manageable.