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

1. Problem Description

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

First, we find the least common multiple of the coefficients 18 and

2

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

$24 = 2^3 \cdot 3$

The LCM of 18 and 24 is $2^3 \cdot 3^2 = 8 \cdot 9 = 72$.

Next, we find the highest power of the term $(2x+y)$ that appears in either denominator.

The first denominator has $(2x+y)^4$, and the second has $(2x+y)^2$. The highest power is $(2x+y)^4$.

Finally, we find the highest power of the term $(x-1)$ that appears in either denominator.

The first denominator has $(x-1)^1$, and the second has $(x-1)^3$. The highest power is $(x-1)^3$.

Therefore, the LCD is the product of the LCM of the coefficients and the highest powers of the terms $(2x+y)$ and $(x-1)$.

LCD $= 72(2x+y)^4(x-1)^3$.

3. Final Answer

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