multiplying rational expressions

Multiplying rational expressions follows the same rule you already use for ordinary fractions: top times top, bottom times bottom. Just as 2/3 · 4/5 = 8/15, you have P/Q · R/S = (P·R)/(Q·S). No common denominator is needed for multiplication.

The smart way to do it, though, is to factor everything first and cancel before you multiply, rather than expanding a big product and untangling it afterward. Cancelling early keeps the numbers small and the algebra clean. So you factor each numerator and denominator, cross-cancel any matching factors, and only then write the result.

Keep an eye on excluded values throughout. Every factor that ever sat in a denominator — even one you cancel — contributes a restriction, and those restrictions carry into the final answer. The product is only valid where all the original pieces were defined.