We are asked to find the least common denominator (LCD) of the following rational expressions: $\frac{-2.5}{16x+32}$, $\frac{13}{6x^2(x+2)^5}$, and $\frac{17}{4x^3(x^2-4)}$.

1. Problem Description

We are asked to find the least common denominator (LCD) of the following rational expressions:

$\frac{-2.5}{16x+32}$, $\frac{13}{6x^2(x+2)^5}$, and $\frac{17}{4x^3(x^2-4)}$.

2. Solution Steps

First, we factor the denominators:

$16x+32 = 16(x+2)$

$6x^2(x+2)^5$ is already factored.

$4x^3(x^2-4) = 4x^3(x-2)(x+2)$

Now we identify the distinct factors: $16, 6, 4, x, (x+2), (x-2)$.

Next, we find the LCD.

The LCM of the coefficients 16, 6, and 4 is

4

8. The highest power of $x$ is $x^3$.

The highest power of $(x+2)$ is $(x+2)^5$.

The highest power of $(x-2)$ is $(x-2)$.

Therefore, the least common denominator (LCD) is

$48x^3(x+2)^5(x-2)$.

3. Final Answer

The least common denominator is $48x^3(x+2)^5(x-2)$.