Solution 1.3:4b

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Current revision (13:58, 22 September 2008) (edit) (undo)
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The numbers
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The numbers 9 and 27 can both be written as powers of 3,
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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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{{Displayed math||<math>\begin{align}
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9 &= 3\cdot 3 = 3^{2}\,,\\[5pt]
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27 &= 3\cdot 9 = 3\cdot 3\cdot 3 = 3^{3}\textrm{.}
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\end{align}</math>}}
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<math>\begin{align}
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Thus, all factors in the expression can be written using a common base and the whole product can be simplified using the power rules
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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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{{Displayed math||<math>\begin{align}
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Thus, all factors in the expression can be written using a common base
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3^{13}\cdot 9^{-3}\cdot 27^{-2} &= 3^{13}\cdot (3^{2})^{-3}\cdot (3^{3})^{-2}\\[3pt]
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&= 3^{13}\cdot 3^{2\cdot (-3)}\cdot 3^{3\cdot (-2)}\\[3pt]
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and the whole product can be simplified using the power rules
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&= 3^{13}\cdot 3^{-6}\cdot 3^{-6}\\[3pt]
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&= 3^{13-6-6}\\[3pt]
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&= 3^{1}\\[3pt]
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<math>\begin{align}
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&= 3\,\textrm{.}
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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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\end{align}</math>}}
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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>
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Current revision

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

\displaystyle \begin{align}

9 &= 3\cdot 3 = 3^{2}\,,\\[5pt] 27 &= 3\cdot 9 = 3\cdot 3\cdot 3 = 3^{3}\textrm{.} \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}\cdot 9^{-3}\cdot 27^{-2} &= 3^{13}\cdot (3^{2})^{-3}\cdot (3^{3})^{-2}\\[3pt] &= 3^{13}\cdot 3^{2\cdot (-3)}\cdot 3^{3\cdot (-2)}\\[3pt] &= 3^{13}\cdot 3^{-6}\cdot 3^{-6}\\[3pt] &= 3^{13-6-6}\\[3pt] &= 3^{1}\\[3pt] &= 3\,\textrm{.} \end{align}

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