Lösung 3.1:7a
Aus Online Mathematik Brückenkurs 1
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First, we multiply the tops and bottoms of the two terms by the conjugate of their respective denominators, so that there are no root signs left in the denominators, | First, we multiply the tops and bottoms of the two terms by the conjugate of their respective denominators, so that there are no root signs left in the denominators, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
\frac{1}{\sqrt{6}-\sqrt{5}} | \frac{1}{\sqrt{6}-\sqrt{5}} | ||
&= \frac{1}{\sqrt{6}-\sqrt{5}}\cdot \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}\\[5pt] | &= \frac{1}{\sqrt{6}-\sqrt{5}}\cdot \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}\\[5pt] | ||
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Now, we can subtract the terms and simplify the result, | Now, we can subtract the terms and simplify the result, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
\frac{1}{\sqrt{6}-\sqrt{5}}-\frac{1}{\sqrt{7}-\sqrt{6}} | \frac{1}{\sqrt{6}-\sqrt{5}}-\frac{1}{\sqrt{7}-\sqrt{6}} | ||
&= \sqrt{6}+\sqrt{5}-(\sqrt{7}+\sqrt{6})\\[5pt] | &= \sqrt{6}+\sqrt{5}-(\sqrt{7}+\sqrt{6})\\[5pt] | ||
Version vom 08:39, 22. Okt. 2008
First, we multiply the tops and bottoms of the two terms by the conjugate of their respective denominators, so that there are no root signs left in the denominators,
| \displaystyle \begin{align}
\frac{1}{\sqrt{6}-\sqrt{5}} &= \frac{1}{\sqrt{6}-\sqrt{5}}\cdot \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}\\[5pt] &= \frac{\sqrt{6}+\sqrt{5}}{(\sqrt{6})^{2}-(\sqrt{5})^{2}}\\[5pt] &= \frac{\sqrt{6}+\sqrt{5}}{6-5}\\[5pt] &= \sqrt{6}+\sqrt{5}\,,\\[10pt] \frac{1}{\sqrt{7}-\sqrt{6}} &= \frac{1}{\sqrt{7}-\sqrt{6}}\cdot \frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}\\[5pt] &= \frac{\sqrt{7}+\sqrt{6}}{(\sqrt{7})^{2}-(\sqrt{6})^{2}}\\[5pt] &= \frac{\sqrt{7}+\sqrt{6}}{7-6}\\[5pt] &= \sqrt{7}+\sqrt{6}\,\textrm{.} \end{align} |
Now, we can subtract the terms and simplify the result,
| \displaystyle \begin{align}
\frac{1}{\sqrt{6}-\sqrt{5}}-\frac{1}{\sqrt{7}-\sqrt{6}} &= \sqrt{6}+\sqrt{5}-(\sqrt{7}+\sqrt{6})\\[5pt] &= \sqrt{6}+\sqrt{5}-\sqrt{7}-\sqrt{6}\\[5pt] &= \sqrt{5}-\sqrt{7}\,\textrm{.} \end{align} |
