I am given a relation that $3X^3 = 2Y^2$
I then have to find the ratio of $X : Y$
To me, the first steps would be the following:
Divide by two: ${3X^3\over2} = {2Y^2\over2}$
${3\over2}X^3 = Y^2$
Then square root: $\sqrt{{3\over2}X^3} = \sqrt{Y^2}$
$\sqrt{{3\over2}X^3} = Y$
This seems pretty ugly now and still includes a cubed power for X. Is this the final reduction or is there a way to get this to the format of $X : Y$?
On
This is an odd question, because the ratio $X:Y$ can be $r : 1$ for any non-negative number $r.$
You can express the ratio in terms of $X$ as in another answer, or you can take the relationship $3X^3 = 2Y^2$ and divide both sides by $3Y^3$:
$$ \frac{X^3}{Y^3} = \frac 2{3Y}.$$
Now take the cube root:
$$ \frac XY = \sqrt[3]{\frac 2{3Y}}. $$
An advantage of this particular formula is that it tells you when when the ratio is positive (namely, when $Y$ is positive) and when the ratio is negative.
You didn't find the ratio though. One way to do this is to note that if $Y = 0$ then the ratio is not defined. Otherwise, $$ \frac{X^2}{Y^2} = \frac{2}{3X} \implies \frac{X}{Y} = \pm \sqrt{\frac{2}{3X}} $$