Basic division problem: dividing a fraction by a fraction I thought I clearly knew how to divide fractions by fractions until I came across this problem. Please can somebody let me know where I am going wrong?

Here is what we start with.
$(1-2x)/(2x^1/2)/e^x$
It then becomes $(1-2x)/(2e^x)*(x^1/2)$
I thought it would end up simplifying to $(e^x - 2xe^x)/(2x^1/2)$

As noted already in a comment, the division bar on the bottom is slightly longer than the one on top, suggesting that the intended grouping of the terms is 
$$ \frac{\left( \frac{1-2x}{2\sqrt{x}} \right)}{e^x},$$
since it is a typographic convention to make the horizontal bar of the "main" fraction longer than the horizontal bars that might occur in the main fraction's numerator or denominator.
It is also a typographic convention that the horizontal bar of the main fraction should line up with the $=$ sign when the fraction set equal to something else in an equation. That is another clue that the numerator of the "main" fraction is 
$\frac{1-2x}{2\sqrt{x}}$ and not just $1-2x.$
A third clue presumably is that what you had before the "combine the numerator" step was consistent with a result of $\frac{1-2x}{2\sqrt{x}}$ in the numerator and $e^x$ in the denominator and not consistent with getting $1-2x$ in the numerator and
$\frac{2\sqrt{x}}{e^x}$ in the denominator.
On the other hand, the picture illustrates some bad typography which likely confused you. First, the tiny difference in lengths of the horizontal bars is subtle and is not really a good thing for a writer to rely upon. Parentheses or some other way of clarifying the fraction's structure would have been better.
Second, the horizontal bar on top is thicker than the one on the bottom, which is bad typesetting and might trick you into thinking the thicker bar is the "main" fraction bar.
Third, the symbols in the fraction in the "main" fraction's numerator are printed the same size as the symbols in the denominator, which robs you of another clue that better typesetting would have provided.
As an alternative to putting parentheses around the entire numerator, the expression could have been written as
$$ \frac{(1-2x)/(2\sqrt{x})}{e^x}$$
or
$$ \frac{ \frac{1}{2\sqrt{x}}(1-2x) }{e^x}$$
or even
$$ \frac{ \frac12 x^{-1/2}(1-2x) }{e^x}.$$
Whether any of those would have been a good idea would depend on what came before the $=$ sign. (And possibly those things could have been better typeset too.) 
For reference, this is how LaTeX or MathJax would typeset an expression written like the one in your first equation:
$$= \frac{ \frac{1-2x}{2\sqrt{x}} }{e^x} \qquad\text{Combine numerator}$$
In addition to lining up the main fraction bar with the $=$ sign and making the main fraction bar slightly longer than the one in the numerator (two things that the typography in the picture in the question got right),
the top bar is not thicker than the bottom bar,
and the symbols in the numerator are slightly smaller than the symbols in the denominator because the numerator is itself a fraction whereas the denominator is not.
In conclusion, it's important to make your intention clear when writing a formula, and for printed material good typesetting is important.
