The problem asks to find the least common denominator (LCD) of the two rational expressions: $\frac{7x}{18(2x+y)^4(x-1)}$ and $\frac{5}{24(2x+y)^2(x-1)^3}$.

1. Problem Description

The problem asks to find the least common denominator (LCD) of the two rational expressions:

$\frac{7x}{18(2x+y)^4(x-1)}$ and $\frac{5}{24(2x+y)^2(x-1)^3}$.

2. Solution Steps

To find the LCD of the two rational expressions, we need to find the least common multiple (LCM) of their denominators.

The denominators are $18(2x+y)^4(x-1)$ and $24(2x+y)^2(x-1)^3$.

First, we find the LCM of the coefficients 18 and

2

4. The prime factorization of 18 is $2 \times 3^2$.

The prime factorization of 24 is $2^3 \times 3$.

Therefore, the LCM of 18 and 24 is $2^3 \times 3^2 = 8 \times 9 = 72$.

Next, we find the LCM of the variable expressions.

For $(2x+y)^4$ and $(2x+y)^2$, the LCM is $(2x+y)^4$ (choose the highest power).

For $(x-1)$ and $(x-1)^3$, the LCM is $(x-1)^3$ (choose the highest power).

Therefore, the LCM of the denominators is $72(2x+y)^4(x-1)^3$.

3. Final Answer

The LCD of the two rational expressions is $72(2x+y)^4(x-1)^3$.

So the answer is b.