Question

20．3${\;}^{\frac{1}{3}}$+$\frac{i}{（{3}^{\frac{1}{3}}-i）^{3}}$=$\frac{10+10•{3}^{\frac{1}{3}}+6•{3}^{\frac{2}{3}}}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}$$+\frac{3-3•{3}^{\frac{1}{3}}}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}i$．

Answer 1

分析 直接利用复数代数形式的乘除运算化简求值．

解答 解：3${\;}^{\frac{1}{3}}$+$\frac{i}{（{3}^{\frac{1}{3}}-i）^{3}}$=${3}^{\frac{1}{3}}+\frac{i}{3-{3}^{\frac{4}{3}}-（{3}^{\frac{5}{3}}-1）i}$
=${3}^{\frac{1}{3}}+\frac{i[（3-{3}^{\frac{4}{3}}）+（{3}^{\frac{5}{3}}-1）i]}{[（3-{3}^{\frac{4}{3}}）-（{3}^{\frac{5}{3}}-1）i][（3-{3}^{\frac{4}{3}}）+（{3}^{\frac{5}{3}}-1）i]}$
=${3}^{\frac{1}{3}}+\frac{（1-{3}^{\frac{5}{3}}）+（3-{3}^{\frac{4}{3}}）i}{（3-{3}^{\frac{4}{3}}）^{2}+（{3}^{\frac{5}{3}}-1）^{2}}$
=${3}^{\frac{1}{3}}+\frac{（1-3•{3}^{\frac{2}{3}}）+3（1-{3}^{\frac{1}{3}}）i}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}$
=$\frac{10+10•{3}^{\frac{1}{3}}+6•{3}^{\frac{2}{3}}}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}$$+\frac{3-3•{3}^{\frac{1}{3}}}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}i$．
故答案为：$\frac{10+10•{3}^{\frac{1}{3}}+6•{3}^{\frac{2}{3}}}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}$$+\frac{3-3•{3}^{\frac{1}{3}}}{10+9•{3}^{\frac{1}{3}}+3•{3}^{\frac{2}{3}}}i$．

点评 本题考查了复数代数形式的乘除运算，考查了计算能力，是基础题．