Problem 49
Question
Find the LCD of each pair of fractions. (lesson \(5-6\) ) $$\frac{1}{3 n}, \frac{7}{6 n^{3}}$$
Step-by-Step Solution
Verified Answer
The LCD is \(6n^3\).
1Step 1: Identify the Denominators
Identify the denominators of the fractions, which are \(3n\) and \(6n^3\).
2Step 2: Factor the Denominators
Factor the denominators into their prime factors: \(3n = 3 \times n\) and \(6n^3 = 2 \times 3 \times n^3\).
3Step 3: Determine the Greatest Powers of Each Factor
Identify the greatest power of each prime factor across the factored denominators: the factors are 2, 3, and \(n\). The greatest powers are 2 (from \(6n^3\)), 3 (from both denominators), and \(n^3\) (from \(6n^3\)).
4Step 4: Calculate the Least Common Denominator (LCD)
Multiply the greatest powers of all prime factors to find the LCD: \[ \text{LCD} = 2^1 \times 3^1 \times n^3 = 6n^3 \]
Key Concepts
Understanding FractionsImportant Role of FactoringExplaining Prime FactorsUnderstanding Denominators
Understanding Fractions
Fractions are a way of representing parts of a whole. They consist of a numerator and a denominator. The numerator is the top number, indicating how many parts we have. The denominator, the bottom part, tells us into how many equal parts the whole is divided.
For example, in the fraction \( \frac{1}{3n} \), 1 is the numerator, and \( 3n \) is the denominator. Similarly, in \( \frac{7}{6n^3} \), 7 is the numerator, and \( 6n^3 \) is the denominator. Understanding fractions is essential for solving problems involving the Least Common Denominator (LCD).
When working with fractions, finding a common denominator helps us compare, add, or subtract them. The least common denominator is the smallest common multiple of the denominators of two or more fractions. This ensures we can work with the fractions more easily and accurately.
For example, in the fraction \( \frac{1}{3n} \), 1 is the numerator, and \( 3n \) is the denominator. Similarly, in \( \frac{7}{6n^3} \), 7 is the numerator, and \( 6n^3 \) is the denominator. Understanding fractions is essential for solving problems involving the Least Common Denominator (LCD).
When working with fractions, finding a common denominator helps us compare, add, or subtract them. The least common denominator is the smallest common multiple of the denominators of two or more fractions. This ensures we can work with the fractions more easily and accurately.
Important Role of Factoring
Factoring is breaking down a number or expression into its constituent parts, or factors, that multiply together to give the original number. In the context of fractions and denominators, factoring is useful for finding the least common denominator.
Consider the denominators \( 3n \) and \( 6n^3 \). They can be factored into prime numbers and variables, which makes it easier to understand their multiple relationships. Factoring helps identify the greatest powers of each factor to compute the LCD.
The denominator \( 3n \) factors into \( 3 \times n \), while \( 6n^3 \) factors into \( 2 \times 3 \times n^3 \). Factoring is a key step in breaking down complex expressions, helping you to solve equations more easily.
Consider the denominators \( 3n \) and \( 6n^3 \). They can be factored into prime numbers and variables, which makes it easier to understand their multiple relationships. Factoring helps identify the greatest powers of each factor to compute the LCD.
The denominator \( 3n \) factors into \( 3 \times n \), while \( 6n^3 \) factors into \( 2 \times 3 \times n^3 \). Factoring is a key step in breaking down complex expressions, helping you to solve equations more easily.
Explaining Prime Factors
Prime factors are the building blocks of numbers, and they are crucial when factoring denominators.A prime number has only two positive divisors: one and itself. To find the prime factors of a number, you break it down until you're only left with prime numbers.
In our exercise, the prime factors are part of the solution. The denominator \( 3n \) has the prime factor 3, whereas \( 6n^3 \) has the prime factors 2 and 3. The variable \( n \) is treated similarly to a prime number in factoring when considering its powers. Identifying prime factors is essential in ensuring that we find the correct least common denominator by considering the greatest powers of each prime factor from all denominators involved.
In our exercise, the prime factors are part of the solution. The denominator \( 3n \) has the prime factor 3, whereas \( 6n^3 \) has the prime factors 2 and 3. The variable \( n \) is treated similarly to a prime number in factoring when considering its powers. Identifying prime factors is essential in ensuring that we find the correct least common denominator by considering the greatest powers of each prime factor from all denominators involved.
Understanding Denominators
Denominators are crucial in fractions as they show how many parts the whole is divided into. When dealing with multiple fractions, finding a common denominator allows for smooth operations like addition and subtraction.
In the given example, the fractions \( \frac{1}{3n} \) and \( \frac{7}{6n^3} \) have denominators \( 3n \) and \( 6n^3 \). Finding the least common denominator (LCD) involves identifying each denominator's prime factors and determining the greatest power of each factor.\[\text{To find the LCD, you combine these factors at their highest powers:}\]
In the given example, the fractions \( \frac{1}{3n} \) and \( \frac{7}{6n^3} \) have denominators \( 3n \) and \( 6n^3 \). Finding the least common denominator (LCD) involves identifying each denominator's prime factors and determining the greatest power of each factor.\[\text{To find the LCD, you combine these factors at their highest powers:}\]
- The factor 2 appears in \( 6n^3 \).
- The factor 3 appears in both \( 3n \) and \( 6n^3 \).
- The variable \( n \) reaches its greatest power, \( n^3 \), in \( 6n^3 \).
Other exercises in this chapter
Problem 49
Find each sum or difference. Write in simplest form. $$\frac{3}{5}+\frac{1}{3}$$
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Evaluate each expression. $$m \div n \text { if } m=-\frac{8}{9} \text { and } n=\frac{7}{18}$$
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The table shows the number of sports films created with different themes. Which theme occurs \(\frac{5}{12}\) as many times as boxing? $$\begin{array}{|l|c|}\hl
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Write a subtraction expression in which the difference of two fractions is \(\frac{18}{25}.\)
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