The problem is to find the Least Common Denominator (LCD) of two rational expressions: $\frac{7x}{18(2x+y)^4(x-1)}$ and $\frac{5}{24(2x+y)^2(x-1)^3}$. We are given four possible answers to choose from.

1. Problem Description

The problem is to find the Least Common Denominator (LCD) of two rational expressions: $\frac{7x}{18(2x+y)^4(x-1)}$ and $\frac{5}{24(2x+y)^2(x-1)^3}$. We are given four possible answers to choose from.

2. Solution Steps

To find the LCD of two rational expressions, we need to find the least common multiple (LCM) of the denominators. The denominators are $18(2x+y)^4(x-1)$ and $24(2x+y)^2(x-1)^3$.

First, let's find the LCM of the coefficients 18 and

2

4. $18 = 2 \times 3^2$

$24 = 2^3 \times 3$

LCM(18, 24) = $2^3 \times 3^2 = 8 \times 9 = 72$

Next, we find the LCM of the variable expressions.

For $(2x+y)^4$ and $(2x+y)^2$, the LCM is $(2x+y)^{\max(4,2)} = (2x+y)^4$.

For $(x-1)$ and $(x-1)^3$, the LCM is $(x-1)^{\max(1,3)} = (x-1)^3$.

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

3. Final Answer

The LCD is $72(2x+y)^4(x-1)^3$. The correct answer is (b).