When working with regular fractions, multiplying and dividing is much easier than adding and subtracting. The sames rules apply for simplifying rational expressions. Lets take a look at an example:
Our first step is to completely factor the numerator and denominator. The first numerator factors to (x-2)(x-6) and the first denominator is a perfect square so that factors into (x+4)(x-4). The second numerator factors into 4(x+4) and the second denominator factors into (x-2)(x-2). This leaves you with (x-2)(x-6)/(x+4)(x-4) times 4(x+4)/(x-2)(x-2).
The next step is to cancel out all the terms that are the same, like this:
You can only cancel out terms that are exactly the same like (x-2) on top and ONE of the (x-2)s on the bottom. One you have canceled all things the same on the top and bottom you combine what you have left as your final answer. In the numerator you were left with 4 and (x-6) and in the denominator you were left with (x-4) and (x-2). You put all of hat together like the last section shown above and you have your final answer. Much easier than it looks.
Just remember to can not cancel TERMS only FACTORS. Meaning if we go back to our original equation you can not just cancel x2 in the numerator and denominator. You must factor first.
When dividing rational expressions you do the same exact thing you do when multiplying. The key to dividing is to remember Keep. Flip. Change. Keep the first set the same, flip the second set, and change the operation to multiplication. Then you factor and cancel out like you normally would.
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