Evaluate:
1(216)−23÷1(27)−43\dfrac{1}{(216)^{-\dfrac{2}{3}}} \div \dfrac{1}{(27)^{-\dfrac{4}{3}}}(216)−321÷(27)−341
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1(216)−23÷1(27)−43=21623÷2743=(63)23÷(33)43=(6)2×33÷(3)4×33=62÷34=3681=49\dfrac{1}{(216)^{-\dfrac{2}{3}}} \div \dfrac{1}{(27)^{-\dfrac{4}{3}}}\\[1em] = 216^{\dfrac{2}{3}} \div 27^{\dfrac{4}{3}}\\[1em] = (6^3)^{\dfrac{2}{3}} \div (3^3)^{\dfrac{4}{3}}\\[1em] = (6)^{\dfrac{2 \times 3}{3}} \div (3)^{\dfrac{4 \times 3}{3}}\\[1em] = 6^2 \div 3^4\\[1em] = \dfrac{36}{81}\\[1em] = \dfrac{4}{9}(216)−321÷(27)−341=21632÷2734=(63)32÷(33)34=(6)32×3÷(3)34×3=62÷34=8136=94
Hence, 1(216)−23÷1(27)−43=49\dfrac{1}{(216)^{-\dfrac{2}{3}}} \div \dfrac{1}{(27)^{-\dfrac{4}{3}}} = \dfrac{4}{9}(216)−321÷(27)−341=94.
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