Final answer:
The least common denominator for the expression (3x)/(x+1)+(x+1)/(2x)+(5)/(x) is 2x(x+1). You find the LCD by determining the least common multiple of the denominators, which in this case involves the factors x and (x+1), multiplied by the prime factor 2.
Step-by-step explanation:
The student is asking about finding the least common denominator (LCD) for the expression (3x)/(x+1)+(x+1)/(2x)+(5)/(x). To find the LCD, we look at each denominator and determine the least common multiple among them. In this case, the denominators are x+1, 2x, and x. The LCD will need to include each of these factors in a way that allows us to combine the fractions together.
The least common multiple of the denominators x+1, 2x, and x can be found by taking the product of the distinct prime factors raised to the highest power they occur in any of the denominators. In this case, the prime factorization of 2x is 2 and x. Since the other denominators include the x and x+1 terms, the least common denominator for the given expression is 2x(x+1).
Once we have the LCD, we can write each fraction with the common denominator and combine them accordingly. For example, if we had fractions ½ and ⅓, their common denominator would be 6, and we would rewrite them as 3/6 and 2/6 before adding the numerators. This ensures that we can combine the fractions into a single fraction with the correct least common denominator.