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Math Help - evaluate

  1. #1
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    evaluate

    without the use of a calculator, evaluate (\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 (\sqrt[3]{0.216})(243^{-\frac{2}{3}}) and leave your answer in fraction.
    Hi Punch,

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

    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,

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

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

    Unfortunately, 243 doesn't work out so neatly. But 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: . \left(\sqrt[3]{0.216}\right)\left({\color{red}343}^{-\frac{2}{3}}\right)

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


    (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: . \frac{3}{5} \times \frac{1}{49} \;=\;\frac{3}{245}

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  5. #5
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    Otherwise, we only have

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

    =\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

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

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