1. ## evaluate

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

2. Originally Posted by Punch
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,

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

$\displaystyle \sqrt[3]{\frac{0.216}{(243)^2}}$

3. 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?

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

5. 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

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

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