A number which is both a perfect square and a perfect cube is a
perfect sixth power:



Any perfect 6th power is a product of perfect 6th powers.  So to make a 
perfect sixth power out of 324 by multiplying it by something,
we have to multiply it by 2 enough times to make it contain a factor
, and we have to also multiply it by 3 enough times to make it 
contain 

So we have to multiply it by 2 four times, which amounts to multiplying
it by , and we also have to multiply it by 3 two times, which 
amounts to multiplying it by .

So we multiply 324 by 

When we multiply 324 by 144 we get 46656 which is both  as well as .

So the smallest possible answer is 144.  
  
There are other answers too, but 144 is the smallest.

The next larger integer we could have used is to multiply 324 by 2 enough 
times to make it a perfect twelfth power, which would be to multuiply it by



When we multiply 324 by 9216 we get 2985984 which is both  as well as .

There are actually an infinite number of correct answers, but I think
your teacher meant just the smallest answer, which is 144.

Edwin