Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Finding the Least Common Denominator (LCD) of Rational Expressions**
When working with rational expressions, finding the Least Common Denominator (LCD) is crucial for addition and subtraction. The LCD is the smallest expression that each of the denominators can divide into without leaving a remainder. To find the LCD, we need to determine the least common multiple (LCM) of the denominators' factors.
**Given:**
The factorizations of the denominators of two rational expressions are as follows:
\[ 5 \cdot 7 \cdot a \cdot a \]
\[ 5 \cdot 7 \cdot 7 \cdot a \cdot a \]
### Steps to Find the LCD:
1. **List all the unique factors:**
- From the factorizations, the unique factors are 5, 7, and \(a\).
2. **Use the highest power of each factor:**
- The highest power of 5 is \(5^1\).
- The highest power of 7 is \(7^2\).
- The highest power of \(a\) is \(a^2\).
3. **Multiply these factors together:**
- \(LCD = 5^1 \cdot 7^2 \cdot a^2\)
- \(LCD = 5 \cdot 49 \cdot a^2\)
- \(LCD = 5 \cdot 49 \cdot a^2\)
- \(LCD = 245a^2\)
Therefore, the LCD of the given rational expressions is \(245a^2\).
**Answer:**
\[ \boxed{245a^2} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc60d0879-bf1d-4600-af9d-cdd728ad041f%2Fbfc8e77e-3ca3-45f7-bcd2-b8014da5a783%2Fd0gazwq_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Least Common Denominator (LCD) of Rational Expressions**
When working with rational expressions, finding the Least Common Denominator (LCD) is crucial for addition and subtraction. The LCD is the smallest expression that each of the denominators can divide into without leaving a remainder. To find the LCD, we need to determine the least common multiple (LCM) of the denominators' factors.
**Given:**
The factorizations of the denominators of two rational expressions are as follows:
\[ 5 \cdot 7 \cdot a \cdot a \]
\[ 5 \cdot 7 \cdot 7 \cdot a \cdot a \]
### Steps to Find the LCD:
1. **List all the unique factors:**
- From the factorizations, the unique factors are 5, 7, and \(a\).
2. **Use the highest power of each factor:**
- The highest power of 5 is \(5^1\).
- The highest power of 7 is \(7^2\).
- The highest power of \(a\) is \(a^2\).
3. **Multiply these factors together:**
- \(LCD = 5^1 \cdot 7^2 \cdot a^2\)
- \(LCD = 5 \cdot 49 \cdot a^2\)
- \(LCD = 5 \cdot 49 \cdot a^2\)
- \(LCD = 245a^2\)
Therefore, the LCD of the given rational expressions is \(245a^2\).
**Answer:**
\[ \boxed{245a^2} \]
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