Say you have an proportion such that
a is proportional to b^2 * c^4. If one doubles a, but leaves c constant, how much does b change by? This is just a sample question, but is there a rule that I can use to find the relation between these three? For example, if c doubles as well as a, how does b change? This is just an example, it can be used for any formula. Here's an example problem:
let a be proportional to b^2 times c^4.
If c is multiplied by 2, but a is halved, how does b change (as in what b is multiplied by?)
In the scenario you're asking about, you're dealing with variables in direct proportion to one another, which simply means that if you increase one, then the other also increases (as opposed to inverse proportion, where an increase in one variable results in a decrease of the other). So we may conclude that increasing $a$ causes an increase in $b$, but by how much? For clarity, rename $c^4$ as $k$. Then we have $a=kb^2$, so $b=\sqrt{\frac{a}{k}}=\frac{\sqrt{a}}{c^2}$. Thus, doubling $a$ causes $b$ to increase by a factor of $\sqrt{2}$.
That's the general idea - just solve the equation for the variable you're interested in and ask yourself, how does it change when one or more of the terms on the other side changes? If you've got your variable equal to a fraction, it's helpful to know that an increase in the numerator will increase the value of your variable and that an increase in the denominator will decrease its value (and vice-versa).
As for what would happen if $c$ doubled as well as $a$, simply substitute $2c$ for $c$ in the above equation. You'll get that $b=\frac{\sqrt{2a}}{4c^2}$, so doubling $c$ leads to a reduction of $b$ by a factor of $4$.