You are correct when you say that raising a product to an nth power is the same as multiplying that product by itself n times. The result of this is that the power gets "distributed" to each factor. That is, in general,
Another useful property is
again with and .
Edit: But be sure you see that we cannot distribute powers over addition or subtraction. That is, for arbitrary and (though there are specific values for which this will be true). In this case, we would have to multiply by itself times.