Here's a less exponent intensive method. Since $a$ and $b$ are positive integers, it follows that $a^2$ is a perfect square and $b^3$ is a perfect cube. Thus, we want to write $432$ as a perfect square times a perfect cube. Having done this, we then multiply the square root of the perfect square (i.e. $a$) by the cube root of the perfect cube (i.e. $b$).
Perfect squares (omitting $1$) are $4,$ $9,$ $16,$ $25,$ etc. If you check to see if $432$ is divisible by $4$ (motivated because $432$ is even; assured by the 4 divisibility rule), you'll find it is, with $432 = 4 \cdot 108.$
Since $108$ is not a perfect cube ($108$ isn't one of $8,$ $27,$ $64,$ $125,$ etc.), there must be a larger perfect square factor of $432$ than $4,$ or equivalently, there must be a perfect square factor of $108.$ Checking for divisibility by $4$ (motivated because $108$ is even; assured by the 4 divisibility rule), we find that $108 = 4 \cdot 27.$ Therefore, from $432 = 4 \cdot 108$ and $108 = 4 \cdot 27,$ we get:
$$432 \;= \;4 \cdot (4 \cdot 27) \;= \;16 \cdot 27$$
Now that we have $432$ written as a perfect square times a perfect cube, it's easy to see that $a = 4$ and $b = 3,$ and so $ab = 4 \cdot 3 = 12.$ [Note to others: In some of the statements above I've assumed the item is sound.]