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受欢迎的代数 >

展开 (z^2-3/z)^{10}

解答

展开

(z^{2} - \frac{3}{z})^{10}

解答

z^{20} - 30 z^{17} + 405 z^{14} - 3240 z^{11} + 17010 z^{8} - 61236 z^{5} + 153090 z^{2} - \frac{262440}{z} + \frac{295245}{z ^{4}} - \frac{196830}{z ^{7}} + \frac{59049}{z ^{10}}

求解步骤

(z^{2} - \frac{3}{z})^{10}

使用二项式定理:

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

a = z^{2}, b = - \frac{3}{z}

= i = 0 \sum 10 (i 10) (z^{2})^{(10 - i)} (- \frac{3}{z})^{i}

展开求和

= \frac{10 !}{0 ! ( 10 - 0 ) !} (z^{2})^{10} (- \frac{3}{z})^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} (z^{2})^{9} (- \frac{3}{z})^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} (z^{2})^{8} (- \frac{3}{z})^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} (z^{2})^{7} (- \frac{3}{z})^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} (z^{2})^{6} (- \frac{3}{z})^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} (z^{2})^{5} (- \frac{3}{z})^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} (z^{2})^{4} (- \frac{3}{z})^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} (z^{2})^{3} (- \frac{3}{z})^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} (z^{2})^{2} (- \frac{3}{z})^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} (z^{2})^{1} (- \frac{3}{z})^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} (z^{2})^{0} (- \frac{3}{z})^{10}

化简

\frac{10 !}{0 ! ( 10 - 0 ) !} (z^{2})^{10} (- \frac{3}{z})^{0} : z^{20}

化简

\frac{10 !}{1 ! ( 10 - 1 ) !} (z^{2})^{9} (- \frac{3}{z})^{1} : - 30 z^{17}

化简

\frac{10 !}{2 ! ( 10 - 2 ) !} (z^{2})^{8} (- \frac{3}{z})^{2} : 405 z^{14}

化简

\frac{10 !}{3 ! ( 10 - 3 ) !} (z^{2})^{7} (- \frac{3}{z})^{3} : - 3240 z^{11}

化简

\frac{10 !}{4 ! ( 10 - 4 ) !} (z^{2})^{6} (- \frac{3}{z})^{4} : 17010 z^{8}

化简

\frac{10 !}{5 ! ( 10 - 5 ) !} (z^{2})^{5} (- \frac{3}{z})^{5} : - 61236 z^{5}

化简

\frac{10 !}{6 ! ( 10 - 6 ) !} (z^{2})^{4} (- \frac{3}{z})^{6} : 153090 z^{2}

化简

\frac{10 !}{7 ! ( 10 - 7 ) !} (z^{2})^{3} (- \frac{3}{z})^{7} : - \frac{262440}{z}

化简

\frac{10 !}{8 ! ( 10 - 8 ) !} (z^{2})^{2} (- \frac{3}{z})^{8} : \frac{295245}{z ^{4}}

化简

\frac{10 !}{9 ! ( 10 - 9 ) !} (z^{2})^{1} (- \frac{3}{z})^{9} : - \frac{196830}{z ^{7}}

化简

\frac{10 !}{10 ! ( 10 - 10 ) !} (z^{2})^{0} (- \frac{3}{z})^{10} : \frac{59049}{z ^{10}}

= z^{20} - 30 z^{17} + 405 z^{14} - 3240 z^{11} + 17010 z^{8} - 61236 z^{5} + 153090 z^{2} - \frac{262440}{z} + \frac{295245}{z ^{4}} - \frac{196830}{z ^{7}} + \frac{59049}{z ^{10}}

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