Solution 2.1:7c

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Current revision (12:10, 23 September 2008) (edit) (undo)
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We multiply the top and bottom of the first term by
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We multiply the top and bottom of the first term by <math>a+1</math>, so that both terms then have the same denominator,
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<math>a+1</math>, so that both terms then have the same
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denominator,
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{{Displayed math||<math>\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{.}</math>}}
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<math>\frac{ax}{a+1}\centerdot \frac{a+1}{a+1}-\frac{ax^{2}}{\left( a+1 \right)^{2}}=\frac{ax\left( a+1 \right)-ax^{2}}{\left( a+1 \right)^{2}}</math>
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Because both terms in the numerator contain the factor <math>ax</math>, we take out that factor, obtaining
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{{Displayed math||<math>\frac{ax(a+1-x)}{(a+1)^{2}}</math>}}
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Because both terms in the numerator contain the factor
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<math>ax</math>, we take out that factor, obtaining
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<math>\frac{ax\left( a+1-x \right)}{\left( a+1 \right)^{2}}</math>
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and see that the answer cannot be simplified any further.
and see that the answer cannot be simplified any further.
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NOTE: It is only factors in the numerator and denominator that can cancel each other out, not
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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
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individual terms. Hence, the following "cancellation" is wrong:
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<math>\frac{ax\left( a+1-x \right)}{\left( a+1 \right)^{2}}=\frac{ax-ax^{2}}{a+1}</math>
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{{Displayed math||<math>\frac{ax(\rlap{/\,/\,/\,/}a+1)-ax^2}{(a+1)^{2\llap{/}}} = \frac{ax-ax^{2}}{a+1}\,\textrm{.}</math>}}

Current revision

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{.}
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