Find the LCD for the following rational expressions.
$\frac{7}{15 x^6}, \frac{4}{25 x^3}$
LCD = $\square$ (Simplify your answer.)
Answer (1)
Find the LCM of the coefficients 15 and 25, which is 75.
Find the LCM of the variable parts x 6 and x 3 , which is x 6 .
Multiply the LCM of the coefficients and the LCM of the variable parts.
The LCD is 75 x 6 .
Explanation
Understanding the Problem We are given two rational expressions: 15 x 6 7 and 25 x 3 4 . Our goal is to find the least common denominator (LCD) of these two expressions. The LCD is the least common multiple (LCM) of the denominators.
Identifying the Denominators The denominators of the given rational expressions are 15 x 6 and 25 x 3 . To find the LCD, we need to find the LCM of these two terms.
Finding the LCM of the Coefficients First, let's find the LCM of the coefficients, 15 and 25. The prime factorization of 15 is 3 × 5 , and the prime factorization of 25 is 5 2 . The LCM of 15 and 25 is 3 × 5 2 = 3 × 25 = 75 .
Finding the LCM of the Variables Next, let's find the LCM of the variable parts, x 6 and x 3 . The LCM of x 6 and x 3 is x 6 because x 6 is divisible by x 3 .
Determining the LCD Now, we multiply the LCM of the coefficients and the LCM of the variable parts to get the LCD of the rational expressions. So, the LCD is 75 x 6 .
Final Answer Therefore, the LCD for the given rational expressions is 75 x 6 .
Examples
When adding or subtracting fractions with different denominators, you need to find the least common denominator (LCD). For example, if you want to add 15 x 6 7 and 25 x 3 4 , you first need to find the LCD, which is 75 x 6 . Then, you rewrite each fraction with the LCD as the denominator: 15 x 6 7 = 15 x 6 × 5 7 × 5 = 75 x 6 35 and 25 x 3 4 = 25 x 3 × 3 x 3 4 × 3 x 3 = 75 x 6 12 x 3 . Now you can add the fractions: 75 x 6 35 + 75 x 6 12 x 3 = 75 x 6 35 + 12 x 3 . This process is essential in various algebraic manipulations and simplifications.
