Exercise 19

The quotient of \( \left( \cfrac{-8}{9} \right)^3 \div \left( \cfrac{12}{27} \right)^3 \) equals to:

  1. \( \left( \cfrac{24}{81} \right)^3 \)

  2. \( -8 \)

  3. \( \cfrac{24}{81} \)

  4. \( 8 \)

  5. \( (-2)^6 \)

Try to solve it before checking the answer.

  1. \( -8 \)

\[
\begin{aligned}
&\left( \frac{-8}{9} \right)^3 \div \left( \frac{12}{27} \right)^3
\\[1em]
&\hspace{5em}=
\left( \frac{-8}{9} \div \frac{12}{27} \right)^3
\\[1em]
&\hspace{5em}=
\left( \frac{-8}{9} \times \frac{27}{12} \right)^3
\\[1em]
&\hspace{5em}=
\left( \frac{ (-1) \times 2^3 \times 3^3 }{ 3^2 \times 2^2 \times 3 } \right)^3
\\[1em]
&\hspace{5em}=
\left( \frac{ (-1) \times 2^3 \times 3^3 }{ 2^2 \times 3^3 } \right)^3
\\[1em]
&\hspace{5em}=
\left( (-1) \times 2 \right)^3
\\[1em]
&\hspace{5em}=
(-1)^3 \times 2^3
\\[1em]
&\hspace{5em}= \boldsymbol{-8}
\end{aligned}
\]
\[
\begin{aligned}
\left( \frac{-8}{9} \right)^3 \div \left( \frac{12}{27} \right)^3
&=
\left( \frac{-8}{9} \div \frac{12}{27} \right)^3
\\[1em]
&=
\left( \frac{-8}{9} \times \frac{27}{12} \right)^3
\\[1em]
&=
\left( \frac{ (-1) \times 2^3 \times 3^3 }{ 3^2 \times 2^2 \times 3 } \right)^3
\\[1em]
&=
\left( \frac{ (-1) \times 2^3 \times 3^3 }{ 2^2 \times 3^3 } \right)^3
\\[1em]
&=
\left( (-1) \times 2 \right)^3
\\[1em]
&=
(-1)^3 \times 2^3
\\[1em]
&= \boldsymbol{-8}
\end{aligned}
\]

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