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  1. #1
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    evaluate

    without the use of a calculator, evaluate$\displaystyle (\sqrt[3]{0.216})(243^{-\frac{2}{3}})$ and leave your answer in fraction.
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  2. #2
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    Quote Originally Posted by Punch View Post
    without the use of a calculator, evaluate$\displaystyle (\sqrt[3]{0.216})(243^{-\frac{2}{3}})$ and leave your answer in fraction.
    Hi Punch,

    $\displaystyle 243^{-\frac{1}{3}}=\frac{1}{(243)^{\frac{1}{3}}}=\left(\ frac{1}{243}\right)^{\frac{1}{3}}$

    $\displaystyle 243^{-\frac{2}{3}}=\left(\frac{1}{(243)^{\frac{1}{3}}}\r ight)^2=\left(\frac{1}{(243)^2}\right)^{\frac{1}{3 }}=\sqrt[3]{\left(\frac{1}{(243)^2}\right)}$

    Hence,

    $\displaystyle \left(\sqrt[3]{0.216}\right)\left(243^{-\frac{2}{3}}\right)=\sqrt[3]{0.216}\ \sqrt[3]{\frac{1}{(243)^2}}$

    $\displaystyle \sqrt[3]{\frac{0.216}{(243)^2}}$
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  3. #3
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    But you can do better than that! $\displaystyle 216= 6^3$ so $\displaystyle 0.216= \frac{216}{1000}= \frac{6^3}{10^3}= (.6)^3$

    Unfortunately, 243 doesn't work out so neatly. But $\displaystyle 7^3= 343$. Are you sure you haven't miscopied?
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  4. #4
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    hELLO, Punch!

    I'll assume that HallsofIvy is right.


    Without the use of a calculator, evaluate: .$\displaystyle \left(\sqrt[3]{0.216}\right)\left({\color{red}343}^{-\frac{2}{3}}\right)$

    $\displaystyle \sqrt[3]{0.216} \;=\;\sqrt[3]{\frac{216}{1000}} \;=\; \sqrt[3]{\frac{6^3}{10^3}} \;=\;\frac{6}{10}\;=\;\frac{3}{5}$


    $\displaystyle (343)^{-\frac{2}{3}} \;=\;\frac{1}{343^{\frac{2}{3}}} \;=\;\frac{1}{(7^3)^{\frac{2}{3}}} \;=\; \frac{1}{7^2} \;=\;\frac{1}{49}$


    Therefore: . $\displaystyle \frac{3}{5} \times \frac{1}{49} \;=\;\frac{3}{245}$

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  5. #5
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    Otherwise, we only have

    $\displaystyle \sqrt[3]{\frac{6^3}{10^3(3)81}}=\frac{6}{10}\sqrt[3]{\frac{1}{(3)3^4}}$

    $\displaystyle =\frac{3}{5}\sqrt[3]{\frac{1}{3^33^2}}=\frac{1}{3}\ \frac{3}{5}\sqrt[3]{\frac{1}{9}}=\frac{1}{5}\sqrt[3]{\frac{1}{9}}$

    or

    $\displaystyle \sqrt[3]{\frac{6^3}{10^33(81)}}=\sqrt[3]{\frac{3(2)3(2)3(2)}{10(10)10(3)3^4}}$

    $\displaystyle =\sqrt[3]{\frac{1}{5(5)5(3)3}}=\frac{1}{5}\sqrt[3]{\frac{1}{9}}$
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