Find the least common denominator (LCD) of the following rational expressions: $\frac{7x}{18(2x+y)^4(x-1)}$ and $\frac{5}{24(2x+y)^2(x-1)^3}$

1. Problem Description

Find the least common denominator (LCD) of the following 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 (LCM) of the 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. $18 = 2 \times 3^2$

$24 = 2^3 \times 3$

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 $(2x+y)^{\max(4,2)} = (2x+y)^4$.

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 answer is b.