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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?

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    $\begingroup$ In your expression, there are 6 symbols and 5 operators. In the other expression, there are 3 symbols and 2 operators (depending a bit on how you count). However, personally I'd prefer the minus sign before the fraction, not in the numerator. (You can even see that the area occupied on the paper is greater in your expression!) In addition, x and a both appear twice in your expression, and you have to think for a while to figure out how the value of the expression changes if x or a is increased (or decreased). In the other expression, this is obvious. $\endgroup$ – Andreas Rejbrand Jan 7 '18 at 21:08
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    $\begingroup$ LeafCount[(1/x - 1/a)/(x - a)]==17>8==LeafCount[-1/(x a)] $\endgroup$ – AccidentalFourierTransform Jan 7 '18 at 21:14
  • $\begingroup$ The first expression is better at showing where the expression comes from and what it means: a difference quotient to be used to find a derivative. The second algebraically equivalent expression is better (because simpler) when you want to evaluate the expression for $x$ close to $a$. So "better" depends on context. $\endgroup$ – Ethan Bolker Jan 8 '18 at 0:48
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    $\begingroup$ @AccidentalFourierTransform it might help to include some explanation, that this is, Mathematica (I think) code and that leaf count (I think) will be a measure for the size/complexity of the term, or something like this. $\endgroup$ – quid Jan 8 '18 at 1:34
  • $\begingroup$ @AndreasRejbrand: "personally I'd prefer the minus sign before the fraction, not in the numerator" - note that that is definitely a personal thing; I prefer the minus sign in the numerator, because if it's next to the fraction it can be too visually similar to the fraction line and so be easily missed (especially if you have handwriting like mine ;-) ) $\endgroup$ – psmears Jan 8 '18 at 8:29
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"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.

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    $\begingroup$ A formula that is written usually with compound fractions is the formula of the Harmonic Mean. $$H=\frac n {\frac 1{x_1}+\cdots +\frac 1 {x_n} }$$ The representation of $H$ without compound statements is much more complex. A simple representation that does express $H$ implicitly and avoids compound fractions is $$\frac n H=\frac 1{x_1}+\cdots +\frac 1 {x_n} $$ $\endgroup$ – miracle173 Jan 8 '18 at 0:33
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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 $a=0$

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    $\begingroup$ Note that the OP didn't show us the original question but only shows us a pair of arithmetical expressions. This isn't even a sentence. So we can only imagine what is the original question. In the context of difference quotients one always assumes that x and a are different are valid arguments of the function f. So I think your objection is meaningless, If it is not clear from the question which values are excluded, then on should add this to the answer. $\endgroup$ – miracle173 Jan 7 '18 at 23:25
  • $\begingroup$ Thanks for the comment.I agree that the question was not well posed . normally we assume that mathematical expressions are meaningful. $\endgroup$ – Mohammad Riazi-Kermani Jan 8 '18 at 0:04
  • $\begingroup$ this was the question... I think it was an exercise about functions. $\endgroup$ – Jwan622 Jan 8 '18 at 4:06
  • $\begingroup$ This. Teacher asks sloppy question, then tries to penalize student for an answer that is technically correct. $\endgroup$ – Federico Poloni Jan 8 '18 at 9:51
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    $\begingroup$ Exactly my point. We need clarification to have equality. $\endgroup$ – Mohammad Riazi-Kermani Jan 9 '18 at 2:15
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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.

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