Simplifying rational expressions means the same as simplifying the fraction. This means that we’ll concentrate on the same terms in the denominator and numerator and try to adjust whole expression, using factoring knowledge we have, in order to simplify given rational expression.
All these tasks can be solved step by step, or you could stop, look at the task and think about it, which can shorten the time of solving and make it more interesting. In the following task we’ll show both ways.
Example 1. Simplify $\frac{3 \cdot a^2}{9 \cdot a \cdot b}$
First thing we notice here is that we only have two terms which is great, because we don’t have to worry if we can shorten something or not- everything is bound with multiplication. If we look, we can see that we have two real numbers which are divisible. So we can shorten those first.
$\frac{3 \cdot a^2}{9 \cdot a \cdot b} = \frac{a^2}{3 \cdot a \cdot b}$
Now we can notice that we have a square $ a^2 = a * a$. We also have “a” in denominator so we can shorten those also. Basically, you are looking what denominator and numerator have in common.
$\frac{a \cdot a}{3 \cdot a \cdot b} = \frac{a}{3 \cdot b}$
Here we come to an end because we don’t know what a and b are.
Second way we could solve this is to look at this expression. First extract all the common factors numerator and denominator have and then simply cross them out.
The common factor of these two terms is $ 3 \cdot a$.
$\frac{3 \cdot a^2}{9 \cdot a \cdot b} = \frac{3 \cdot a \cdot a}{3 \cdot 3 \cdot a \cdot b} = \frac{a}{3b}$
In most tasks you won’t get factored expressions right away, it will be your job to factor it. For this you can use all the tricks you learned including usual factoring, the difference of square, square of binomial, cube of binomial and so on.
Example 2. Simplify $\frac{3a + 3b}{2a + 2b}$
If it is hard for you to see what the common term of denominator and numerator is, you can first factor them separately and see if anything comes up.
We can see that from the numerator we can extract 3 and from the denominator 2. We get the same term in the brackets which is equal to 1.
$\frac{3a + 3b}{2a + 2b} = \frac{3(a + b)}{2(a + b)} = \frac{3}{2}$
Example 3. Simplify $\frac{12x^2y – 18xy^2}{12x^2y – 8x^3}$
This task is a bit messy, so you should take your time and solve it step by step starting with extracting the common terms from numerator and denominator.
$\frac{12x^2y – 18xy^2}{12x^2y – 8x^3} = \frac{6xy * (2x – 3y)}{4x^2(3y – 2x)}$
Now, on the first look, there isn’t much to be done here, we can only simplify “2x”. But if you look closely, these two brackets are the same only with different sign. If we subtract -1 from one of them we can make them equal.
$\frac{6xy \cdot (2x – 3y)}{4x^2(3y – 2x)} = \frac{-3y \cdot (-2x + 3y)}{2x \cdot (3y – 2x)}$
Since we can use commutative property of subtraction and addition we can do the following:
$\frac{-3y \cdot (-2x + 3y)}{2x \cdot (3y – 2x)} = \frac{-3y \cdot (3y – 2x)}{2x \cdot (3y – 2x)} = – \frac{3y}{2x}$
Example 4. Simplify $\frac{a^2 – (b + c)^2}{(a + b)^2 – c^2}$
Don’t go rushing in and expand these binomials, take a look at what you have. We have difference of squares which we can easily factorize.
$\frac{a^2 – (b + c)^2}{(a + b)^2 – c^2} = \frac{(a – b – c)(a + b + c)}{(a + b – c)(a + b + c)} = \frac{a – b – c}{a + b – c}$
