[tex]\cfrac{512^{\frac{1}{3}}}{64^{\frac{2}{3}}}\implies \cfrac{(2^9)^{\frac{1}{3}}}{(2^6)^{\frac{2}{3}}}\implies \cfrac{2^3}{2^4}\implies \cfrac{1}{2^4\cdot 2^{-3}}\implies \cfrac{1}{2} \\\\[-0.35em] ~\dotfill\\\\ \boxed{A}\qquad \left( \cfrac{64^{\frac{2}{3}}}{512^{\frac{1}{3}}} \right)^{-1}\implies \left( \cfrac{512^{\frac{1}{3}}}{64^{\frac{2}{3}}} \right)^{+1}\implies \stackrel{from~above}{\cfrac{1}{2}} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\boxed{B}\qquad \left( 64^{\frac{2}{3}}\times 512^{-\frac{1}{3}} \right)^{-1}\implies \left( \cfrac{64^{\frac{2}{3}}}{512^{\frac{1}{3}}} \right)^{-1}\implies \left( \cfrac{512^{\frac{1}{3}}}{64^{\frac{2}{3}}} \right)^{+1}\implies \stackrel{from~above}{\cfrac{1}{2}} \\\\[-0.35em] ~\dotfill\\\\ \boxed{C}\qquad \cfrac{64^{-\frac{2}{3}}}{512^{-\frac{1}{3}}}\implies \cfrac{512^{\frac{1}{3}}}{64^{\frac{2}{3}}}\implies\stackrel{from~above}{\cfrac{1}{2}}[/tex]
