You thought you were knee-deep in complexity up to now? Brace yourself. Welcome to the wacky world of complex rational expressions. These guys have fractions inside their fractions. They may drive you nuts if you think about them too long. Don't worry—you can simplify the complex. Just treat 'em like long division.
Sample Problem
Simplify 
.
Looks scary? Chill. It's only long division. Rewrite it as a division problem.
Now flip and multiply.
Cancel like factors.
Which gives us:
Sample Problem
Simplify 
.
Flip and switch to multiplication.
Factor and cancel.
And the answer is:
2(x – 1)
Sample Problem
Simplify 
.
Before we flip the switch and proceed to multiplication, we gotta do the subtraction and addition in the numerator and denominator.
The numerator:
The denominator:
Now, rewrite the whole thing as a multiplication problem.
Factor the a2 – 1:
Cancel like factors.
That gives us the final answer, along with a well-earned victory dance:
a – 1