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受欢迎的代数 >

展开 ((x^2)/y-(y^{1/2})/x)^9

解答

展开

(\frac{x ^{2}}{y} - \frac{y ^{\frac{1}{2}}}{x})^{9}

解答

\frac{x ^{18}}{y ^{9}} - \frac{9 x ^{15}}{y ^{\frac{15}{2}}} + \frac{36 x ^{12}}{y ^{6}} - \frac{84 x ^{9}}{y ^{\frac{9}{2}}} + \frac{126 x ^{6}}{y ^{3}} - \frac{126 x ^{3}}{y ^{\frac{3}{2}}} + 84 - \frac{36 y ^{\frac{3}{2}}}{x ^{3}} + \frac{9 y ^{3}}{x ^{6}} - \frac{y ^{\frac{9}{2}}}{x ^{9}}

求解步骤

(\frac{x ^{2}}{y} - \frac{y ^{\frac{1}{2}}}{x})^{9}

使用二项式定理:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = \frac{x ^{2}}{y}, b = - \frac{y ^{\frac{1}{2}}}{x}

= i = 0 \sum 9 (i 9) (\frac{x ^{2}}{y})^{(9 - i)} (- \frac{y ^{\frac{1}{2}}}{x})^{i}

展开求和

= \frac{9 !}{0 ! ( 9 - 0 ) !} (\frac{x ^{2}}{y})^{9} (- \frac{y ^{\frac{1}{2}}}{x})^{0} + \frac{9 !}{1 ! ( 9 - 1 ) !} (\frac{x ^{2}}{y})^{8} (- \frac{y ^{\frac{1}{2}}}{x})^{1} + \frac{9 !}{2 ! ( 9 - 2 ) !} (\frac{x ^{2}}{y})^{7} (- \frac{y ^{\frac{1}{2}}}{x})^{2} + \frac{9 !}{3 ! ( 9 - 3 ) !} (\frac{x ^{2}}{y})^{6} (- \frac{y ^{\frac{1}{2}}}{x})^{3} + \frac{9 !}{4 ! ( 9 - 4 ) !} (\frac{x ^{2}}{y})^{5} (- \frac{y ^{\frac{1}{2}}}{x})^{4} + \frac{9 !}{5 ! ( 9 - 5 ) !} (\frac{x ^{2}}{y})^{4} (- \frac{y ^{\frac{1}{2}}}{x})^{5} + \frac{9 !}{6 ! ( 9 - 6 ) !} (\frac{x ^{2}}{y})^{3} (- \frac{y ^{\frac{1}{2}}}{x})^{6} + \frac{9 !}{7 ! ( 9 - 7 ) !} (\frac{x ^{2}}{y})^{2} (- \frac{y ^{\frac{1}{2}}}{x})^{7} + \frac{9 !}{8 ! ( 9 - 8 ) !} (\frac{x ^{2}}{y})^{1} (- \frac{y ^{\frac{1}{2}}}{x})^{8} + \frac{9 !}{9 ! ( 9 - 9 ) !} (\frac{x ^{2}}{y})^{0} (- \frac{y ^{\frac{1}{2}}}{x})^{9}

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} (\frac{x ^{2}}{y})^{9} (- \frac{y ^{\frac{1}{2}}}{x})^{0} : \frac{x ^{18}}{y ^{9}}

化简

\frac{9 !}{1 ! ( 9 - 1 ) !} (\frac{x ^{2}}{y})^{8} (- \frac{y ^{\frac{1}{2}}}{x})^{1} : - \frac{9 x ^{15}}{y ^{\frac{15}{2}}}

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} (\frac{x ^{2}}{y})^{7} (- \frac{y ^{\frac{1}{2}}}{x})^{2} : \frac{36 x ^{12}}{y ^{6}}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} (\frac{x ^{2}}{y})^{6} (- \frac{y ^{\frac{1}{2}}}{x})^{3} : - \frac{84 x ^{9}}{y ^{\frac{9}{2}}}

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} (\frac{x ^{2}}{y})^{5} (- \frac{y ^{\frac{1}{2}}}{x})^{4} : \frac{126 x ^{6}}{y ^{3}}

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} (\frac{x ^{2}}{y})^{4} (- \frac{y ^{\frac{1}{2}}}{x})^{5} : - \frac{126 x ^{3}}{y ^{\frac{3}{2}}}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} (\frac{x ^{2}}{y})^{3} (- \frac{y ^{\frac{1}{2}}}{x})^{6} : 84

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} (\frac{x ^{2}}{y})^{2} (- \frac{y ^{\frac{1}{2}}}{x})^{7} : - \frac{36 y ^{\frac{3}{2}}}{x ^{3}}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} (\frac{x ^{2}}{y})^{1} (- \frac{y ^{\frac{1}{2}}}{x})^{8} : \frac{9 y ^{3}}{x ^{6}}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} (\frac{x ^{2}}{y})^{0} (- \frac{y ^{\frac{1}{2}}}{x})^{9} : - \frac{y ^{\frac{9}{2}}}{x ^{9}}

= \frac{x ^{18}}{y ^{9}} - \frac{9 x ^{15}}{y ^{\frac{15}{2}}} + \frac{36 x ^{12}}{y ^{6}} - \frac{84 x ^{9}}{y ^{\frac{9}{2}}} + \frac{126 x ^{6}}{y ^{3}} - \frac{126 x ^{3}}{y ^{\frac{3}{2}}} + 84 - \frac{36 y ^{\frac{3}{2}}}{x ^{3}} + \frac{9 y ^{3}}{x ^{6}} - \frac{y ^{\frac{9}{2}}}{x ^{9}}

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