Find the LCD for the following rational expressions.
$\frac{8}{5 a^4 b^5}, \frac{9}{10 a^3 b^7}$
LCD = $\square$ (Simplify your answer.)
Answer (1)
Identify the denominators of the rational expressions: 5 a 4 b 5 and 10 a 3 b 7 .
Determine the least common multiple (LCM) of the coefficients 5 and 10, which is 10.
Identify the highest powers of each variable: a 4 and b 7 .
Combine the LCM of the coefficients and the highest powers of the variables to form the LCD: 10 a 4 b 7 .
Explanation
Understanding the Problem We are given two rational expressions: 5 a 4 b 5 8 and 10 a 3 b 7 9 . 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 are 5 a 4 b 5 and 10 a 3 b 7 . To find the LCM, we need to find the LCM of the coefficients and the highest power of each variable present in the denominators.
Finding the LCM of the Coefficients First, let's find the LCM of the coefficients 5 and 10. The multiples of 5 are 5, 10, 15, ... and the multiples of 10 are 10, 20, 30, .... The least common multiple of 5 and 10 is 10.
Finding the Highest Powers of Variables Next, we find the highest power of each variable. For the variable a , we have a 4 and a 3 . The highest power is a 4 . For the variable b , we have b 5 and b 7 . The highest power is b 7 .
Calculating the LCD Now, we multiply the LCM of the coefficients and the highest powers of each variable to get the LCD. So, the LCD is 10 a 4 b 7 .
Final Answer Therefore, the LCD for the given rational expressions is 10 a 4 b 7 .
Examples
When adding or subtracting fractions with polynomial denominators, finding the LCD is essential. For example, if you are combining terms in an electrical circuit calculation or determining the resultant force in a physics problem, you might encounter rational expressions. The LCD helps to simplify these expressions into a single fraction, making further calculations easier. This skill is also crucial in calculus when integrating rational functions using partial fraction decomposition.
