Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.
No jsMath TeX fonts found -- using image fonts instead.
These may be slow and might not print well.
Use the jsMath control panel to get additional information.
jsMath Control PanelHide this Message

jsMath

Lösung 3.1:7a

Aus Online Mathematik Brückenkurs 1

(Unterschied zwischen Versionen)
Wechseln zu: Navigation, Suche
K
Zeile 1: Zeile 1:
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,
-
 
+
{{Displayed math||<math>\begin{align}
-
<math>\begin{align}
+
\frac{1}{\sqrt{6}-\sqrt{5}}
-
& \frac{1}{\sqrt{6}-\sqrt{5}}=\frac{1}{\sqrt{6}-\sqrt{5}}\centerdot \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}=\frac{\sqrt{6}+\sqrt{5}}{\left( \sqrt{6} \right)^{2}-\left( \sqrt{5} \right)^{2}}=\frac{\sqrt{6}+\sqrt{5}}{6-5}=\sqrt{6}+\sqrt{5}, \\
+
&= \frac{1}{\sqrt{6}-\sqrt{5}}\cdot \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}\\[5pt]
-
& \frac{1}{\sqrt{7}-\sqrt{6}}=\frac{1}{\sqrt{7}-\sqrt{6}}\centerdot \frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}=\frac{\sqrt{7}+\sqrt{6}}{\left( \sqrt{7} \right)^{2}-\left( \sqrt{6} \right)^{2}}=\frac{\sqrt{7}+\sqrt{6}}{7-6}=\sqrt{7}+\sqrt{6}, \\
+
&= \frac{\sqrt{6}+\sqrt{5}}{(\sqrt{6})^{2}-(\sqrt{5})^{2}}\\[5pt]
-
& \\
+
&= \frac{\sqrt{6}+\sqrt{5}}{6-5}\\[5pt]
-
\end{align}</math>
+
&= \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}</math>}}
Now, we can subtract the terms and simplify the result,
Now, we can subtract the terms and simplify the result,
-
 
+
{{Displayed math||<math>\begin{align}
-
<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}-\left( \sqrt{7}+\sqrt{6} \right) \\
+
&= \sqrt{6}+\sqrt{5}-(\sqrt{7}+\sqrt{6})\\[5pt]
-
& =\sqrt{6}+\sqrt{5}-\sqrt{7}-\sqrt{6}=\sqrt{5}-\sqrt{7}. \\
+
&= \sqrt{6}+\sqrt{5}-\sqrt{7}-\sqrt{6}\\[5pt]
-
\end{align}</math>
+
&= \sqrt{5}-\sqrt{7}\,\textrm{.}
 +
\end{align}</math>}}

Version vom 14:05, 30. Sep. 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,

Vorlage:Displayed math

Now, we can subtract the terms and simplify the result,

Vorlage:Displayed math

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_3.1:7a“