How do we multiply and divide rational expressions?

When multiplying rational expressions alot of things usually cancel out. For example if you have a/b times b/a the a and b would both cancel. When the a cancel each other out that would become one. when the b cancel it that would also become one. you are now left with 1 times 1 so your final answer would be 1. A could not equal to 0 and neither can b due to the fact that 0 can not be in the denominator. When dividing rational expressions the operation sign changes and the second fraction flips. For example say if your problem is 1/x divided by 1/2x , you would keep the first fraction the way it is. You when then change the division sign to a multiplication sign and flip the 1/2x. Your new problem is now  1/x times 2x/1. You multiply straight across and get 2x/x. the x's cancel each other and your final answer is 2.