Problem 12 - Algebra 1: Concepts and Skills

Question

Find the least common denominator of the pair of rational expressions. $$ \frac{10}{13 v^{7}}, \frac{10}{3 v^{5}} $$

Step-by-Step Solution

Answer

The least common denominator for the pair of rational expressions is \(39v^{7}\).

Step 1: Identify the Denominators

The first rational expression has a denominator of \(13v^{7}\) while the second one has a denominator of \(3v^{5}\).

Step 2: Calculate the LCM of Numerical Parts of the Denominators

The numerical parts of the denominators are 13 and 3. Their least common multiple (LCM) can be found by multiplying 13 and 3, which equals 39.

Step 3: Calculate the LCD of Variables in the Denominators

The variables in the denominators are \(v^{7}\) and \(v^{5}\). The greatest power of v that appears in the denominators is 7, so we take \(v^{7}\).

Step 4: Form the LCD

To form the least common denominator, we multiply the result in step 2 and the result in step 3. So, \(39 * v^{7} = 39v^{7}\) is the LCD.

Key Concepts

Understanding Rational Expressions

Rational expressions can seem daunting at first, but they become straightforward when you break them down. They are simply fractions with polynomials in the numerator and denominator. Just like regular fractions, they can be added, subtracted, multiplied, and divided.
A rational expression looks like this: \(\frac{N}{D}\)where \(N\) and \(D\) are polynomials and \(Deq0\). In the given problem, \(\frac{10}{13v^7}\) and \(\frac{10}{3v^5}\) are rational expressions with the same numerator, making it simpler to see the comparison.When working with rational expressions, especially in finding the least common denominator (LCD), it helps to visualize them as you would numerical fractions. You look for common factors in the denominators, which are crucial in aligning the expressions for addition or subtraction.In essence, rational expressions follow many of the same principles as simpler fractions, with added focus on managing variables and coefficients.

Breaking Down the Numerical Parts

One important step in finding the least common denominator is focusing on the numerical parts of the denominators. They are the constant numbers that you see apart from the variable components. In our problem, these are 13 and 3 from the denominators \(13v^7\) and \(3v^5\).Finding the least common multiple (LCM) of these numbers is simple:
- Identify each number: 13 and 3.- Multiply these numbers together to find their LCM because they are prime and have no common factors.- So, \(13 \times 3 = 39\).The LCM of the numerical parts is crucial for aligning denominators efficiently. By focusing on the numerical parts first, you simplify the process, ensuring the denominators combine accurately in operations involving addition or subtraction.

Greatest Power - A Focus on Variables

When dealing with variables in rational expressions, identifying the greatest power helps streamline finding the LCD. In our example, we focus on the variable \(v\), which appears in both denominators: \(v^7\) and \(v^5\).Here's how we determine which power to use:
- Look at the variable parts of each denominator.- Compare the exponents: 7 from \(v^7\) and 5 from \(v^5\).- Choose the largest exponent as the greatest power, which in this case is 7.This greatest power approach helps ensure that whatever adjustments are made to one expression, the variable part remains uniform across all expressions. By selecting \(v^7\), you effectively align the denominators to accommodate a unified base for any arithmetic operations.