Lösung 2.1:7c
Aus Online Mathematik Brückenkurs 1
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Version vom 13:51, 22. Okt. 2008
We multiply the top and bottom of the first term by \displaystyle a+1, so that both terms then have the same denominator,
| \displaystyle \frac{ax}{a+1}\cdot\frac{a+1}{a+1} - \frac{ax^{2}}{(a+1)^{2}} = \frac{ax(a+1)-ax^{2}}{(a+1)^{2}}\,\textrm{.} |
Because both terms in the numerator contain the factor \displaystyle ax, we take out that factor, obtaining
| \displaystyle \frac{ax(a+1-x)}{(a+1)^{2}} |
and see that the answer cannot be simplified any further.
Note: It is only factors in the numerator and denominator that can cancel each other out, not individual terms. Hence, the following "cancellation" is wrong
| \displaystyle \frac{ax(\rlap{/\,/\,/\,/}a+1)-ax^2}{(a+1)^{2\llap{/}}} = \frac{ax-ax^{2}}{a+1}\,\textrm{.} |
