rationalizing: 2/(x^3 (x - 3)) + 3/(x (x - 3)^2)

[helpmepls!]

this is confusing to me. I want to know how the numerators in the second slide go to the numerators after that. do you factor??

 

[ksdhart2]

Okay, so your attachment is a bit small and hard for me to read, but here's what I think it says:

1)

\(\displaystyle \dfrac{2}{x^3(x-3)}+\dfrac{3}{x(x-3)^2}\)

Denominator 1: x³(x-3)
Denominator 2: x(x-3)²
LCD: x³(x-3)²

\(\displaystyle =\dfrac{2}{x^3(x-3)} \cdot \dfrac{x^3(x-3)^2}{x^3(x-3)^2} + \dfrac{3}{x(x-3)^2} \cdot \dfrac{x^3(x-3)^2}{x^3(x-3)^2}\)

\(\displaystyle =\dfrac{2}{1} \cdot \dfrac{(x-3)}{x^3(x-3)^2} + \dfrac{3}{1} \cdot \dfrac{x^2}{x^3(x-3)^2}\)

Assuming the above is correct, what did you get when you tried simplifying the expression by cancelling like terms? For instance, I see that the numerator and denominator of the left fraction each have an x³ term, and the numerator and the denominator of the right fraction each have an (x-3)² term. What happens if you cancel those? Remember that the goal is to be left with the same denominator in both fractions so you can later add them. You'll sometimes have multiple choices about what terms to cancel, so be careful and be mindful of the end goal.