Solution 1.3:4b

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m (Lösning 1.3:4b moved to Solution 1.3:4b: Robot: moved page)
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The numbers
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<center> [[Image:1_3_4b.gif]] </center>
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<math>9</math>
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and
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<math>27</math>
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can both be written as powers of
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<math>3</math>,
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<math>\begin{align}
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& 9=3\centerdot 3=3^{2} \\
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& \\
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& 27=3\centerdot 9=3\centerdot 3\centerdot 3=3^{3} \\
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\end{align}</math>
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Thus, all factors in the expression can be written using a common base
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and the whole product can be simplified using the power rules
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<math>\begin{align}
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& 3^{13}\centerdot 9^{-3}27^{-2}=3^{13}\centerdot \left( 3^{2} \right)^{-3}\centerdot \left( 3^{3} \right)^{-2} \\
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& \\
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& =3^{13}\centerdot 3^{2\centerdot \left( -3 \right)}\centerdot 3^{3\centerdot \left( -2 \right)}=3^{13}\centerdot 3^{-6}\centerdot 3^{-6} \\
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& \\
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& =3^{13-6-6}=3^{1}=3 \\
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\end{align}</math>

Revision as of 11:48, 15 September 2008

The numbers \displaystyle 9 and \displaystyle 27 can both be written as powers of \displaystyle 3,


\displaystyle \begin{align} & 9=3\centerdot 3=3^{2} \\ & \\ & 27=3\centerdot 9=3\centerdot 3\centerdot 3=3^{3} \\ \end{align}


Thus, all factors in the expression can be written using a common base

and the whole product can be simplified using the power rules


\displaystyle \begin{align} & 3^{13}\centerdot 9^{-3}27^{-2}=3^{13}\centerdot \left( 3^{2} \right)^{-3}\centerdot \left( 3^{3} \right)^{-2} \\ & \\ & =3^{13}\centerdot 3^{2\centerdot \left( -3 \right)}\centerdot 3^{3\centerdot \left( -2 \right)}=3^{13}\centerdot 3^{-6}\centerdot 3^{-6} \\ & \\ & =3^{13-6-6}=3^{1}=3 \\ \end{align}

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