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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Question
![**Problem Statement:**
Find the LCD (Least Common Denominator) of the fractions:
\[
\frac{1}{4x^2y^5} \quad \text{and} \quad \frac{7}{6x^4y^3z^6}
\]
**Explanation:**
To find the Least Common Denominator (LCD) of these fractions, follow these steps:
1. **Factor the Denominators:**
- The first denominator is \(4x^2y^5\). It can be factored into \(2^2 \cdot x^2 \cdot y^5\).
- The second denominator is \(6x^4y^3z^6\). It can be factored into \(2 \cdot 3 \cdot x^4 \cdot y^3 \cdot z^6\).
2. **Identify the Highest Power for Each Factor:**
- For the number 2: The highest power is \(2^2\).
- For the number 3: The power is \(3^1\).
- For the variable \(x\): The highest power is \(x^4\).
- For the variable \(y\): The highest power is \(y^5\).
- For the variable \(z\): The highest power is \(z^6\).
3. **Form the LCD:**
The LCD is the product of the highest powers of all factors that appear in either fraction. Thus, the LCD is:
\[
2^2 \cdot 3^1 \cdot x^4 \cdot y^5 \cdot z^6 = 12x^4y^5z^6
\]
Therefore, the LCD of the fractions \(\frac{1}{4x^2y^5}\) and \(\frac{7}{6x^4y^3z^6}\) is \(12x^4y^5z^6\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbb7c1f4-ae14-42ca-94ec-850570e86a9e%2Fcc2ff76e-07d7-42fd-8e75-4ac97d22351d%2Fh09voq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the LCD (Least Common Denominator) of the fractions:
\[
\frac{1}{4x^2y^5} \quad \text{and} \quad \frac{7}{6x^4y^3z^6}
\]
**Explanation:**
To find the Least Common Denominator (LCD) of these fractions, follow these steps:
1. **Factor the Denominators:**
- The first denominator is \(4x^2y^5\). It can be factored into \(2^2 \cdot x^2 \cdot y^5\).
- The second denominator is \(6x^4y^3z^6\). It can be factored into \(2 \cdot 3 \cdot x^4 \cdot y^3 \cdot z^6\).
2. **Identify the Highest Power for Each Factor:**
- For the number 2: The highest power is \(2^2\).
- For the number 3: The power is \(3^1\).
- For the variable \(x\): The highest power is \(x^4\).
- For the variable \(y\): The highest power is \(y^5\).
- For the variable \(z\): The highest power is \(z^6\).
3. **Form the LCD:**
The LCD is the product of the highest powers of all factors that appear in either fraction. Thus, the LCD is:
\[
2^2 \cdot 3^1 \cdot x^4 \cdot y^5 \cdot z^6 = 12x^4y^5z^6
\]
Therefore, the LCD of the fractions \(\frac{1}{4x^2y^5}\) and \(\frac{7}{6x^4y^3z^6}\) is \(12x^4y^5z^6\).
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