Complex, or "Stacked", Fractions
Purplemath
Once you have learned about rational expressions, and about simplifying, adding/subtracting, and multiplying rational expressions, you'll be faced with dividing rational expressions.
Remembering that fractions are division, the "dividing rational expressions" exercises are commonly formatted as "fractions" where each of the numerator and denominator is itself a rational expression. These expressions are called "complex" fractions.
What is a complex fraction?
A complex fraction is a fraction whose numerator and denominator are themselves rational expressions. That is, they are polynomial fractions whose top and bottom are themselves also, or which also contain, polynomial fractions.
(Note: I don't care for this terminology. I think they use "complex" in this context to stand for "really complicated". But the word "complex", in mathematics, already has a definition; it stands for numbers which include the imaginary number, which is the square root of −1. Fractions with complex numbers in the denominator do have to be simplified [as here], so "complex fractions", being fractions containing complex numbers, are a thing that exist in mathematics; they are a thing with which you may have to deal. But the "complex" fractions under discussion here are not at all the same thing.)
Instead of "complex" fractions, I sometimes refer to these rational-expression fractions as "stacked" fractions, because they tend to have polynomial fractions (that is, rational expressions) stacked on top of each other, as you'll see below.
How do you simplify complex fractions?
There are two methods for simplifying complex fractions:
- Simplify the numerator and denominator separately, convert each to being one fractional expression, flip, multiply, and cancel, if possible.
- Find one LCM for everything, multiply through, and cancel, if possible.
What is the easiest way to simplify complex fractions?
When simplifying complex fractions, the method that is easiest will vary with the student. Some find it easier to deal with the numerator and denominator separately; others prefer multiplying through by the Least Common Multiple (Lowest Common Denominator) of all the fractions in play. I can't tell you which method you'll end up finding more helpful; only you can tell, and only after practice.
The first method starts with finding common denominators for each of the complex numerator and complex denominator in the stacked-fraction expression. Then you convert the complex numerator and complex denominator to their respective common denominators, and combine everything in the complex numerator and in the complex denominator into single fractions. Then, once you've got one fraction (in the complex numerator) divided by another fraction (in the complex denominator), you flip-n-multiply. (Remember that, when you are dividing by a fraction, you flip the bottom fraction and turn the division into multiplication.)
The second method starts with finding one common denominator for all the various denominators in the stacked fraction, multiply top and bottom of the stacked fraction by this one expression, and simplify.
Does the second method sound easier? Yes. Is it actually easier? Most students (including me) would say "yes". Is it easier for *you*? Only you can say for sure.
What is an example of simplifying complex fractions?
- Simplify the following complex fraction:
![[4 + (1/x)] / [3 + (2/x^2)]](rational/compfr01.gif){kind=small}
This complex, or stacked, fraction is formed of two fractional expressions, one on top of the other. To simplify, I'm going to simplify each of the complex numerator and complex denominator separately. Then, once I've got (one fraction) divided by (another fraction), I'll flip-n-multiply, and cancel the common factors, if there are any.
Before I start with the simplification, I note that I can't have x = 0, because this would cause division by zero. This restriction may magically disappear by the time I'm finished simplifying. In order to make sure that my final answer is equal to the original expression, in the sense of having the same domain, I'll need to loop back at the end to check on this restriction.
For the simplification, here are my steps:
![[4 + (1/x)] / [3 + (2/x^2)] = [(4x/x) + (1/x)] / [(3x^2/x^2) + (2/x^2)] = [(4x + 1)/x] / [(3x^2 + 2)/x^2] = [(4x + 1)/x] × [x^2/(3x^2 + 2)] = [(4x + 1)/1] × [x/(3x^2 + 2)] = [(x)(4x + 1)] / [(1)(3x^2 + 2)] = (4x^2 + x)/(3x^2 + 2)](rational/compfr11.png)
Nothing cancels at this point — because terms that are parts of larger expressions don't cancel; only factors cancel, and there are no common factors here — so the last fractional expression above is the final answer.
(The "for x not equal to zero" part is because, in the original expression, "x = 0" would have caused division by zero in the complex fraction, and because this restriction is not apparent in the final expression. Depending on your book and instructor, you may not need to account for this technicality. If you're not sure, ask now, before the test.)
In the exercise above, I demonstrated the flip-n-multiply method of simplification. The other method is to find one common denominator for all the fractions in the expression, and then multiply both the complex numerator and complex denominator by this expression. Then simplify the result.
The two denominators within the numerator and denominator of the stacked fraction are x and x2. The least common denominator is then x2. So I'll multiply, top and bottom, by .
![[4 + (1/x)] / [3 + (2/x^2)] = {[4 + (1/x)] / [3 + (2/x^2)]} × {[x^2/1] / [x^2/1]} = [(4/1)(x^2/1) + (1/x)(x^2/1)] / [(3/1)(x^2/1) + (2/x^2)(x^2/1)] = [(4x^2/1) + (x/1)] / [(3x^2/1) + (2/1)] = [4x^2 + x] / [3x^2 + 2]](rational/compfr12.png)
Then the final answer is:
By multiplying through, top and bottom, by the same thing, I was really just multiplying by 1. This is similar to multiplying the fraction by to convert it to .
In my experience, books and teachers often use the first method for simplifying complex fractions, but students generally prefer the second method. When I was in school, I was taught the first method. As soon as I encountered the second method, I switched to it. In the remaining examples, I will generally demonstrate this second method, but you can use either method you prefer.
- Simplify the following complex fraction:
![[ 1/(x − 1) + x + 3 ] / [ x − 3 + 1/(x + 4) ]](rational/compfr03.gif)
Can I start by hacking off x's? Or lopping off 3's? (Hint: No!) I can only cancel off factors, not terms, so I can't do any cancelling yet. Instead, the first thing I'll do is find the LCM for the fractions in this expression.
The LCM (Least Common Multiple, or, for us older types, the LCD, Lowest Common Denominator) of the given denominators within this complex fraction is (x − 1)(x + 4), so (after noting that x ≠ 1, -4) I'll multiply through, top and bottom, by this expression:
(If you're not sure how I multiplied those factors to get the cubic results, review this lesson on multiplying polynomials.)
Okay; so I've arrived at this expression:
Can I now cancel off the x3's? Or cancel the 6's into the 12? Can I go inside the adding and rip out parts of some of the terms? (Hint: Still no!)
Since only factors cancel, I can try to factor these cubic polynomials. The numerator's cubic factors as (x + 4)(x2 + 2x − 2), while the denominator's cubic factors as (x − 1)(x2 + x − 11). The factored forms share no common factors.
Sharing no factors, nothing cancels. But the quadratic does generate another two zeroes. Once I am aware of all the restrictions, I can state my answer:
(Wondering why the restrictions on the answer? See the explanation here.)
Technically, I probably didn't have to include the domain restrictions of "for x ≠ 1, −4", because these were clear in the original expression. But the x ≠ −4 restriction has disappeared by the end of my computations, and the x ≠ 1 restriction is only apparent if I take the trouble to factor the denominator. So, to be on the safe side, I've included these restrictions from the original stacked-fraction expression.
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