Solution 1.3:4e

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Lösning 1.3:4e moved to Solution 1.3:4e: Robot: moved page)
Line 1: Line 1:
-
{{NAVCONTENT_START}}
+
Because
-
<center> [[Image:1_3_4e.gif]] </center>
+
<math>5^{9}=5^{8+1}=5^{8}\centerdot 5^{1}=5^{8}\centerdot 5</math>,
-
{{NAVCONTENT_STOP}}
+
the two terms inside the brackets have
 +
<math>5^{8}</math>
 +
as a common factor
 +
and can therefore be taken outside the bracket.
 +
 
 +
 
 +
<math>\begin{align}
 +
& \left( 5^{8}+5^{9} \right)^{-1}=\left( 5^{8}+5^{8}\centerdot 5 \right)^{-1}=\left( 5^{8}\centerdot \left( 1+5 \right) \right)^{-1} \\
 +
& \\
 +
& =\left( 5^{8}\centerdot 6 \right)^{-1}=5^{8\centerdot \left( -1 \right)}\centerdot 6^{-1}=5^{-8}\centerdot 6^{-1}. \\
 +
\end{align}</math>
 +
 
 +
Furthermore,
 +
<math>625=5\centerdot 125=5\centerdot 5\centerdot 25=5\centerdot 5\centerdot 5\centerdot 5=5^{4}</math>
 +
and we obtain
 +
 
 +
 
 +
<math>\begin{align}
 +
& 625\centerdot \left( 5^{8}+5^{9} \right)^{-1}=5^{4}\centerdot 5^{-8}\centerdot 6^{-1}=5^{4-8}\centerdot 6^{-1} \\
 +
& \\
 +
& =5^{-4}\centerdot 6^{-1}=\frac{1}{5^{4}}\centerdot \frac{1}{6}=\frac{1}{5^{4}\centerdot 6}=\frac{1}{5\centerdot 5\centerdot 5\centerdot 5\centerdot 6} \\
 +
& \\
 +
& =\frac{1}{3750} \\
 +
\end{align}</math>

Revision as of 11:57, 15 September 2008

Because \displaystyle 5^{9}=5^{8+1}=5^{8}\centerdot 5^{1}=5^{8}\centerdot 5, the two terms inside the brackets have \displaystyle 5^{8} as a common factor and can therefore be taken outside the bracket.


\displaystyle \begin{align} & \left( 5^{8}+5^{9} \right)^{-1}=\left( 5^{8}+5^{8}\centerdot 5 \right)^{-1}=\left( 5^{8}\centerdot \left( 1+5 \right) \right)^{-1} \\ & \\ & =\left( 5^{8}\centerdot 6 \right)^{-1}=5^{8\centerdot \left( -1 \right)}\centerdot 6^{-1}=5^{-8}\centerdot 6^{-1}. \\ \end{align}

Furthermore, \displaystyle 625=5\centerdot 125=5\centerdot 5\centerdot 25=5\centerdot 5\centerdot 5\centerdot 5=5^{4} and we obtain


\displaystyle \begin{align} & 625\centerdot \left( 5^{8}+5^{9} \right)^{-1}=5^{4}\centerdot 5^{-8}\centerdot 6^{-1}=5^{4-8}\centerdot 6^{-1} \\ & \\ & =5^{-4}\centerdot 6^{-1}=\frac{1}{5^{4}}\centerdot \frac{1}{6}=\frac{1}{5^{4}\centerdot 6}=\frac{1}{5\centerdot 5\centerdot 5\centerdot 5\centerdot 6} \\ & \\ & =\frac{1}{3750} \\ \end{align}

Retrieved from "http://wiki.sommarmatte.se/wikis/2008/bridgecourse1-ImperialCollege/index.php/Solution_1.3:4e"