What makes this answer unsimplified?

Question

Generally, what are the rules for simplifying answers?

I had this question:

Given $f(x) = \frac{1}{x}$, evaluate $\frac{f(x) - f(a)}{x-a}$.

Why is the following a bad answer:

$$\frac{\frac{1}{x} - \frac{1}{a}}{x-a}$$

The better answer is:

$$\frac{-1}{xa}$$

Why is this? Is it generally just bad to leave fractions in either the numerator or denominator? So we should try to eliminate the fraction in the numerator by solving the equation first on the top by using a common denominator?

Answer 1

"Is it generally just bad to leave fractions in either the numerator or denominator?"

Yes, exactly. Intuitively, the simplified form of an expression is supposed to be the, well, simplest form. Exactly what constitutes "fully simplified" is subjective and can vary from person to person (is $\sqrt{2}\over 2$ simpler than $1\over\sqrt{2}$?), but some things are pretty much constant across the board, one being that "fractions inside fractions" are to be avoided unless getting rid of them would add significant complexity to the expression.

Answer 2

Suppose you want to compute this for a particular value of $x$ and $a$. In your first answer, you have to perform two divisions, a subtraction, another subtraction, and finally another division. In the second answer, you have to multiply once and divide once. Much easier, no?

It is precisely this decrease in complexity that makes the second answer "more simplified" than the first one.

Answer 3

I like your bad answer better because $$\frac{\frac{1}{x}-\frac{1}{a}}{x-a}=-\frac{1}{ax}$$ is true only if $x\ne a.$

For $x=a$ the left hand side is undefined while the right hand side is defined unless $a=0$