Problem 43 - Elementary and Intermediate Algebra

Question

Find the LCD of each pair of rational expressions. \(\frac{33}{15 a^{3}}, \frac{9}{10 a}\)

Step-by-Step Solution

Answer

The LCD of the expressions is \(30a^3\).

Step 1: Identify the Denominators

Look at the denominators of each rational expression. They are \(15a^3\) and \(10a\).

Step 2: Determine the Prime Factorization of Coefficients

Break down the numerical parts of each denominator. \[15 = 3 imes 5\]\[10 = 2 imes 5\]

Step 3: Identify the Variable Parts

The variable part of the first denominator is \(a^3\) and for the second it is \(a\).

Step 4: Combine the Prime Factors and Variables for the LCD

To find the least common denominator (LCD), take the highest power of all prime numbers and variables that appear in the denominators:- Highest power of 2 is \(2^1\)- Highest power of 3 is \(3^1\)- Highest power of 5 is \(5^1\)- Highest power of \(a\) is \(a^3\)Thus, \[\text{LCD} = 2 imes 3 imes 5 imes a^3 = 30a^3\]

Step 5: Verify the LCD

Verify that each denominator \(15a^3\) and \(10a\) can divide the \(30a^3\):- \(15 = 3 imes 5\) divides \(2 imes 3 imes 5\)- \(10 = 2 imes 5\) divides \(2 imes 3 imes 5\)- \(a^3\) divides \(a^3\); \(a\) divides \(a^3\)

Key Concepts

Understanding Rational Expressions

Rational expressions are similar to fractions but involve polynomials in the numerator and denominator. This means you are working with expressions that look like a fraction but with polynomial components. The denominator cannot be zero, just in the same way the denominator of a fraction cannot be zero. For example, in the exercise, we are dealing with \(\frac{33}{15a^3}\) and \(\frac{9}{10a}\). The key is to handle these expressions carefully, keeping in mind the rules of arithmetic for fractions and adding the layers of polynomial operations.

Exploring Prime Factorization

Prime factorization is crucial for simplifying the process of finding the Least Common Denominator (LCD). It involves breaking down numbers into their "prime" number building blocks, which are numbers greater than 1 that have no divisors other than 1 and themselves. To find the prime factorization of the denominators in our exercise, we break down 15 and 10 as follows:

• 15 becomes \(3 \times 5\).
• 10 becomes \(2 \times 5\).

This step sets the stage for combining these factors, ensuring that we adequately account for all necessary components.

Handling Variables in Denominators

Including variables in denominators requires careful attention. Like numbers, variables also follow rules for determining the LCD. In the exercise, variables are represented as powers of \(a\). It's important to keep track of the highest power of each variable, as it will contribute to the LCD determination:

• The first denominator includes \(a^3\).
• The second has \(a^1\) (simply \(a\)).

To find the LCD, take the highest power of each variable present, which in this case is \(a^3\). This ensures the resulting denominator can accommodate both expressions when they're converted to have the same denominator.