Question

19．化简：$\frac{a-b}{{a}^{\frac{2}{3}}+{a}^{\frac{1}{3}}{b}^{\frac{1}{3}}+{b}^{\frac{2}{3}}}$-$\frac{{a}^{\frac{2}{3}}{-b}^{\frac{2}{3}}}{{a}^{\frac{1}{3}}{-b}^{\frac{1}{3}}}$．

Answer 1

分析 结合分数指数审幂的性质和运算法则利用立方差公式和平方差公式求解．

解答 解：$\frac{a-b}{{a}^{\frac{2}{3}}+{a}^{\frac{1}{3}}{b}^{\frac{1}{3}}+{b}^{\frac{2}{3}}}$-$\frac{{a}^{\frac{2}{3}}{-b}^{\frac{2}{3}}}{{a}^{\frac{1}{3}}{-b}^{\frac{1}{3}}}$
=$\frac{（{a}^{\frac{1}{3}}-{b}^{\frac{1}{3}}）（{a}^{\frac{2}{3}}+{a}^{\frac{1}{3}}{b}^{\frac{1}{3}}+{b}^{\frac{2}{3}}）}{{a}^{\frac{2}{3}}+{a}^{\frac{1}{3}}{b}^{\frac{1}{3}}+{b}^{\frac{2}{3}}}$-$\frac{（{a}^{\frac{1}{3}}-{b}^{\frac{1}{3}}）（{a}^{\frac{1}{3}}+{b}^{\frac{1}{3}}）}{{a}^{\frac{1}{3}}-{b}^{\frac{1}{3}}}$
=（${a}^{\frac{1}{3}}-{b}^{\frac{1}{3}}$）-（${a}^{\frac{1}{3}}+{b}^{\frac{1}{3}}$）
=-2${b}^{\frac{1}{3}}$．

点评 本考查有理数指数幂的化简求值，是基础题，解题时要注意分数指数幂的性质和运算法则的合理运用．