Problem 44 - Elementary and Intermediate Algebra

Question

Find the LCD of each pair of rational expressions. \(\frac{m-21}{12 m^{4}}, \frac{m+1}{18 m}\)

Step-by-Step Solution

Answer

The LCD is \( 36 m^4 \).

Step 1: Identify the Denominators

The given expressions are \( \frac{m-21}{12 m^{4}} \) and \( \frac{m+1}{18 m} \). Each expression has a denominator. The denominators are \( 12 m^{4} \) and \( 18 m \), respectively.

Step 2: Prime Factorize the Coefficients

Prime factorize the numerical coefficients of each denominator. \( 12 = 2^2 \times 3 \) and \( 18 = 2 \times 3^2 \).

Step 3: Factorize the Variables

For the variable parts, we have \( m^4 \) in the first denominator and \( m \) in the second denominator. The factorization of these is simply \( m^4 = m \times m \times m \times m \) and \( m = m \).

Step 4: Determine the Least Common Multiple (LCM)

To find the least common denominator (LCD), take the highest power of each prime factor appearing in the factorizations from Step 2. For the numerical coefficients, we take \( 2^2 \) and \( 3^2 \). For the variable, take the highest power, which is \( m^4 \). Thus, the LCM is \( 2^2 \times 3^2 \times m^4 \).

Step 5: Calculate the LCD

Calculate the product of the factors found in Step 4: \( 2^2 = 4 \), \( 3^2 = 9 \), thus \( 4 \times 9 = 36 \). Then multiply that by the highest power of the variable, giving the least common denominator: \( 36 m^4 \).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of integers, they involve polynomials in the numerator, the denominator, or both. A rational expression is defined as the quotient of two polynomials. For example, in the expressions \( \frac{m-21}{12m^4} \) and \( \frac{m+1}{18m} \), the numerators and the denominators are both polynomials. The primary goal when working with rational expressions is to simplify them or find a common denominator when combining or comparing. Simplifying rational expressions often involves factoring, which reduces them to their simplest form. When working with multiple rational expressions, finding a common denominator is essential because it ensures that the expressions can be combined or compared directly.

Prime Factorization

Prime factorization is the process of expressing a number or polynomial as a product of its prime factors. Prime numbers are numbers that have only two positive divisors: 1 and themselves.

• For example, the number 12 can be broken down into its prime factors: \(12 = 2^2 \times 3\).
• Similarly, 18 becomes \(18 = 2 \times 3^2\).

For the variables in rational expressions, factorization involves expanding them into their simplest multiplicative forms, such as \(m^4 = m \times m \times m \times m\). This process is crucial for finding the Least Common Denominator (LCD) because it allows us to identify all the prime factors we might need to account for. By taking the highest powers of all prime numbers from each denominator, we can construct the LCD effectively.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) of any set of numbers is the smallest multiple that is exactly divisible by each number in the set. For rational expressions, the concept is applied to the denominators. Finding the LCM ensures a unified baseline for calculation or comparison.

• The process involves identifying all distinct prime factors from each number.
• Then, take the highest power of each prime factor present.

For example, given denominators \(12m^4\) and \(18m\), their prime factorization helps determine the LCM. From the prime factors \(2^2 \times 3\) for 12 and \(2 \times 3^2\) for 18, the highest powers are \(2^2\) and \(3^2\). For the variable \(m\), take the highest power, \(m^4\). Therefore, the LCM, and consequently the LCD, is \(36m^4\): multiplying \(4 \times 9\) results in 36, and combining it with \(m^4\) provides the complete LCD.