Solution 1.3:4b

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The numbers 9 and 27 can both be written as powers of 3,
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<center> [[Image:1_3_4b.gif]] </center>
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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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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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{{Displayed math||<math>\begin{align}
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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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&= 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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&= 3\,\textrm{.}
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\end{align}</math>}}

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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