Problem 64 - Algebra A Combined Function

Question

Choose the correct \(L C D\) of \(\frac{5}{14 x^{2}}\) and \(\frac{y}{6 x^{3}}\). a. \(84 x^{5}\) b. \(84 x^{3}\) c. \(42 x^{3}\) d. \(42 x^{5}\)

Step-by-Step Solution

Answer

The correct LCD is option c: \(42 x^{3}\).

Step 1: Identify the Denominators

First, identify the denominators of the two fractions. They are \(14x^2\) and \(6x^3\).

Step 2: Factor the Denominators

Factor each denominator into its prime and variable components. \(14x^2 = 2 \cdot 7 \cdot x^2\) and \(6x^3 = 2 \cdot 3 \cdot x^3\).

Step 3: Determine the Least Common Denominator of Numerical Factors

Find the least common multiple (LCM) of the numerical parts: 14 and 6. The prime factors are 2, 7, 3. The LCM is \(2 \cdot 3 \cdot 7 = 42\).

Step 4: Determine the Least Common Denominator of Variables

For the variable part, choose the highest power of \(x\) present in the denominators. Between \(x^2\) and \(x^3\), the highest power is \(x^3\).

Step 5: Combine Results to Find the LCD

Combine results from Step 3 and Step 4 to find the LCD: \(42 \cdot x^3 = 42x^3\).

Key Concepts

Factoring

Factoring is a method of breaking down numbers or expressions into multiples or components called factors. Factors are numbers or variables that, when multiplied together, give the original number or expression. When we have expressions like in the problem, factoring helps simplify them and identify lowest common denominators (LCDs).
When factoring denominators, such as in the exercise, we look for both numerical factors and variable components.
For example, for the expression "\(14x^2\)", we identified it as \(2 \cdot 7 \cdot x^2\). Similarly, \(6x^3\) was broken down into \(2 \cdot 3 \cdot x^3\).
This process reveals all the single components involved, making it easier to find common ground between different fractions or expressions.

Least Common Multiple (LCM)

The Least Common Multiple, or LCM, is the smallest number that is a multiple of two or more numbers.
When finding the LCM of numbers, we carry out prime factorization to ensure all prime factors are included at least in their highest power that appears in any of the numbers.

• In the provided solution, we sought the LCM of 14 and 6 by looking at their prime factors: 14 (\(2 \cdot 7\)) and 6 (\(2 \cdot 3\)).
• The LCM was determined by multiplying each prime factor the greatest number of times it appears in any of the factorizations: \(2 \cdot 3 \cdot 7 = 42\).

In our context, LCM helps to align the denominators of fractions, providing a basis to add, subtract, or compare fractions effectively and accurately.

Prime Factorization

Prime factorization is the process of expressing a number as the product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
This technique is vital in simplifying and solving problems involving fractions and other complex expressions.
For the problem, each number in the denominators, such as 14 and 6, was broken down into its prime factors:

• 14 becomes \(2 \cdot 7\)
• 6 becomes \(2 \cdot 3\)

Prime factorization ensures we cover each aspect of a number's structure to find common multiples or factors efficiently. It paves the way to solving problems that involve least common denominators.

Rational Expressions

Rational expressions are fractions that include polynomials in the numerator, the denominator, or both. They behave similarly to numeric fractions but include variables.
When working with rational expressions, it's crucial to identify common denominators through least common multiples and factorization.

• In the exercise, we had rational expressions \(\frac{5}{14x^2}\) and \(\frac{y}{6x^3}\).
• The task involved finding a least common denominator (LCD) to potentially align these expressions for additional operations, if necessary.

Understanding rational expressions and simplifying them through techniques like factoring and prime factorization allows for proper arithmetic operations and simplifying complex algebraic fractions. It's a foundational element in algebra that provides tools for managing expressions across diverse mathematical problems.