I'm given a word problem that's as follows:
A, B, C each think of expression that is fraction with $1$ as a numerator and a constant integer times some power of $x$ as a denominator.
The simplest common denominator between A and B expresssions is $4x^{2}$.
The simplest common denominator between B and C expresssions is $12x^{3}$.
The simplest common denominator between A and C expresssions is $6x^{3}$.
Find all possible expressions that could be C's expressions.
How do I even begin to reason about this problem?
Rather than phrasing with fractions... letting $a,b,c$ be the denominators of $A,B,C$ respectively the problem is saying that $\text{lcm}(a,b)=4x^2, \text{lcm}(b,c)=12x^3$ and $\text{lcm}(a,c)=6x^3$.
You should be able to reason that $a$ must be one of $1,2,4,x,2x,4x,x^2,2x^2,4x^2$ since these are the possible factors of $\text{lcm}(a,b)=4x^2$. Similarly for $b$.
Meanwhile, we notice that since $\text{lcm}(b,c)$ and $\text{lcm}(a,c)$ both have $x^3$ in it that $c$ must be a multiple of $x^3$ since had it been a multiple of $x^4$ or higher the $\text{lcm}$ would have been with at least as large an exponent as well while on the other hand had $c$ been a multiple of $x^2$ or lower then the $\text{lcm}$ would not have needed an $x^3$...
Further, we notice that since $\text{lcm}(a,b)$ did not have a factor of $3$ it must be that neither $a$ nor $b$ have a factor of $3$ in them but $\text{lcm}(a,c)$ and $\text{lcm}(b,c)$ both have a factor of $3$, implying that $c$ must have a factor of $3$ present.
Putting all of this together, we find that $a,b,c$ must be one of the following:
$$a=2x^2,b=4x^2,c=3x^3$$
$$a=x^2,b=4x^2,c=6x^3$$
$$a=2x^2,b=4x, c=3x^3$$
$$\vdots$$
and so on... where $b$ is one of $4, 4x, 4x^2$, and $a$ is one of $1,2,x,2x,x^2,2x^2$, and at least one of them being a multiple of $x^2$ while $c$ is either $6x^3$ or $3x^3$.