Find the LCD for the following rational expressions:
[tex]$\frac{5}{6 x^4}, \frac{2}{9 x^2}$[/tex]
Answer (2)
Find the least common multiple (LCM) of the coefficients 6 and 9, which is 18.
Find the LCM of the variable parts x 4 and x 2 , which is x 4 .
Multiply the LCM of the coefficients and the LCM of the variable parts to get the LCD.
The LCD of the given rational expressions is 18 x 4 .
Explanation
Understanding the Problem We are asked to find the Least Common Denominator (LCD) of two rational expressions: 6 x 4 5 and 9 x 2 2 . The LCD is the least common multiple of the denominators.
Identifying the Denominators The denominators are 6 x 4 and 9 x 2 . To find the LCD, we need to find the least common multiple of the coefficients (6 and 9) and the least common multiple of the variable parts ( x 4 and x 2 ).
Finding the LCM of the Coefficients First, let's find the least common multiple (LCM) of the coefficients 6 and 9. The multiples of 6 are 6, 12, 18, 24, ... and the multiples of 9 are 9, 18, 27, ... The least common multiple of 6 and 9 is 18.
Finding the LCM of the Variable Parts Next, let's find the least common multiple of the variable parts x 4 and x 2 . To find the LCM of variables with exponents, we take the highest power of the variable that appears in either expression. In this case, the highest power of x is x 4 .
Determining the LCD Finally, the LCD of the given rational expressions is the product of the LCM of the coefficients and the LCM of the variable parts. Therefore, the LCD is 18 x 4 .
Examples
When adding or subtracting fractions with different denominators, you need to find a common denominator. The LCD is the smallest such common denominator, which simplifies the calculations. For example, if you were combining the work rates of two machines, one working at a rate proportional to 6 x 4 5 and another at 9 x 2 2 , finding the LCD would help you determine their combined work rate efficiently. This concept is also useful in electrical engineering when analyzing circuits with parallel components.
The Least Common Denominator (LCD) of the rational expressions 6 x 4 5 and 9 x 2 2 is found to be 18 x 4 . This is done by finding the least common multiple of the coefficients and the variable parts. Therefore, the LCD is 18 x 4 .
;
