Problem 41 - Elementary and Intermediate Algebra

Question

Find the LCD of each pair of rational expressions. \(\frac{1}{2 x}, \frac{9}{6 x}\)

Step-by-Step Solution

Answer

The LCD of \(\frac{1}{2x}\) and \(\frac{9}{6x}\) is \(6x\).

Step 1: Identify the Denominators

The given rational expressions have the denominators \(2x\) and \(6x\). We need to find a common denominator for these two.

Step 2: Determine the Least Common Multiple (LCM)

To find the LCD, we must determine the least common multiple of the denominators \(2x\) and \(6x\).

Step 3: List the Factors of Each Denominator

The factors of \(2x\) are \(2 \cdot x\). The factors of \(6x\) are \(2 \cdot 3 \cdot x\).

Step 4: Find the LCM of the Numerical Coefficients

The numerical part of \(2x\) is 2, and the numerical part of \(6x\) is 6. The LCM of 2 and 6 is 6.

Step 5: Find the LCM of the Variables

Both denominators have \(x\) as a common variable. The LCM of \(x\) and \(x\) is \(x\).

Step 6: Combine the LCMs

Combine the LCM of the numerical coefficients (6) with the LCM of the variable \(x\) to find the LCD. The LCD is \(6x\).

Key Concepts

Understanding the Least Common Denominator (LCD)

Rational expressions can often have different denominators, which can make calculations cumbersome. To address this, we find the least common denominator (LCD). The LCD helps us to rewrite rational expressions so that they have the same denominator. This simplification allows for easy addition, subtraction, or comparison of rational expressions.

The LCD is essentially the smallest expression into which each of the original denominators can divide without a remainder. It plays a crucial role because operations on rational expressions, similar to fractions, require a common denominator.

• For example, if you have \( \frac{1}{2x} \) and \( \frac{9}{6x} \), their least common denominator is \( 6x \).

Understanding how to find the LCD is fundamental, especially as expressions become more complex. It allows us to explore operations like addition and subtraction with ease.

Exploring the Least Common Multiple (LCM)

The least common multiple, or LCM, is a concept not just confined to numbers but extends to algebraic expressions as well.

The LCM of two or more expressions is the smallest expression that is a multiple of each of them. In the context of rational expressions, finding the LCM of denominators is a key step towards determining the LCD.

• Consider the denominators \( 2x \) and \( 6x \). Their numerical coefficients are 2 and 6.
• The LCM of the numbers 2 and 6 is 6, as 6 is the smallest number divisible by both 2 and 6.
• Similarly, for variables like \( x \), when both expressions have the same variable, it directly becomes part of the LCM.

The LCM gives us a structured path to combine both the coefficients and variables to find the lowest common denominator.

Breaking Down the Factors of Denominators

Finding the factors of each denominator is like understanding the building blocks of algebraic expressions. It gives insight into how these expressions are structured.

For denominators like \( 2x \) and \( 6x \), understanding their factors helps determine both the LCM and ultimately the LCD.

• The factorization of \( 2x \) gives us \( 2 \cdot x \).
• The factorization of \( 6x \) results in \( 2 \cdot 3 \cdot x \).

The key takeaway is to recognize common and unique factors among the denominators.
Gaining proficiency in finding factorization helps in identifying what needs to be multiplied together to find the LCM, and subsequently, the LCD.
Each factor contributes to constructing expressions that denote a common understanding of the denominators.