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展开 (1/2-(sqrt(3))/2 i)^{16}

解答

展开

(\frac{1}{2} - \frac{3}{2} i)^{16}

解答

- \frac{1}{2} + \frac{3}{2} i

求解步骤

(\frac{1}{2} - \frac{3}{2} i)^{16}

使用二项式定理:

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

a = \frac{1}{2}, b = - \frac{3}{2} i

= i = 0 \sum 16 (i 16) (\frac{1}{2})^{(16 - i)} (- \frac{3}{2} i)^{i}

展开求和

= \frac{16 !}{0 ! ( 16 - 0 ) !} (\frac{1}{2})^{16} (- \frac{3}{2} i)^{0} + \frac{16 !}{1 ! ( 16 - 1 ) !} (\frac{1}{2})^{15} (- \frac{3}{2} i)^{1} + \frac{16 !}{2 ! ( 16 - 2 ) !} (\frac{1}{2})^{14} (- \frac{3}{2} i)^{2} + \frac{16 !}{3 ! ( 16 - 3 ) !} (\frac{1}{2})^{13} (- \frac{3}{2} i)^{3} + \frac{16 !}{4 ! ( 16 - 4 ) !} (\frac{1}{2})^{12} (- \frac{3}{2} i)^{4} + \frac{16 !}{5 ! ( 16 - 5 ) !} (\frac{1}{2})^{11} (- \frac{3}{2} i)^{5} + \frac{16 !}{6 ! ( 16 - 6 ) !} (\frac{1}{2})^{10} (- \frac{3}{2} i)^{6} + \frac{16 !}{7 ! ( 16 - 7 ) !} (\frac{1}{2})^{9} (- \frac{3}{2} i)^{7} + \frac{16 !}{8 ! ( 16 - 8 ) !} (\frac{1}{2})^{8} (- \frac{3}{2} i)^{8} + \frac{16 !}{9 ! ( 16 - 9 ) !} (\frac{1}{2})^{7} (- \frac{3}{2} i)^{9} + \frac{16 !}{10 ! ( 16 - 10 ) !} (\frac{1}{2})^{6} (- \frac{3}{2} i)^{10} + \frac{16 !}{11 ! ( 16 - 11 ) !} (\frac{1}{2})^{5} (- \frac{3}{2} i)^{11} + \frac{16 !}{12 ! ( 16 - 12 ) !} (\frac{1}{2})^{4} (- \frac{3}{2} i)^{12} + \frac{16 !}{13 ! ( 16 - 13 ) !} (\frac{1}{2})^{3} (- \frac{3}{2} i)^{13} + \frac{16 !}{14 ! ( 16 - 14 ) !} (\frac{1}{2})^{2} (- \frac{3}{2} i)^{14} + \frac{16 !}{15 ! ( 16 - 15 ) !} (\frac{1}{2})^{1} (- \frac{3}{2} i)^{15} + \frac{16 !}{16 ! ( 16 - 16 ) !} (\frac{1}{2})^{0} (- \frac{3}{2} i)^{16}

化简

\frac{16 !}{0 ! ( 16 - 0 ) !} (\frac{1}{2})^{16} (- \frac{3}{2} i)^{0} : \frac{1}{65536}

化简

\frac{16 !}{1 ! ( 16 - 1 ) !} (\frac{1}{2})^{15} (- \frac{3}{2} i)^{1} : - \frac{3 i}{4096}

化简

\frac{16 !}{2 ! ( 16 - 2 ) !} (\frac{1}{2})^{14} (- \frac{3}{2} i)^{2} : \frac{45 i ^{2}}{8192}

化简

\frac{16 !}{3 ! ( 16 - 3 ) !} (\frac{1}{2})^{13} (- \frac{3}{2} i)^{3} : - \frac{105 3 i ^{3}}{4096}

化简

\frac{16 !}{4 ! ( 16 - 4 ) !} (\frac{1}{2})^{12} (- \frac{3}{2} i)^{4} : \frac{4095 i ^{4}}{16384}

化简

\frac{16 !}{5 ! ( 16 - 5 ) !} (\frac{1}{2})^{11} (- \frac{3}{2} i)^{5} : - \frac{2457 3 i ^{5}}{4096}

化简

\frac{16 !}{6 ! ( 16 - 6 ) !} (\frac{1}{2})^{10} (- \frac{3}{2} i)^{6} : \frac{27027 i ^{6}}{8192}

化简

\frac{16 !}{7 ! ( 16 - 7 ) !} (\frac{1}{2})^{9} (- \frac{3}{2} i)^{7} : - \frac{19305 3 i ^{7}}{4096}

化简

\frac{16 !}{8 ! ( 16 - 8 ) !} (\frac{1}{2})^{8} (- \frac{3}{2} i)^{8} : \frac{521235 i ^{8}}{32768}

化简

\frac{16 !}{9 ! ( 16 - 9 ) !} (\frac{1}{2})^{7} (- \frac{3}{2} i)^{9} : - \frac{57915 3 i ^{9}}{4096}

化简

\frac{16 !}{10 ! ( 16 - 10 ) !} (\frac{1}{2})^{6} (- \frac{3}{2} i)^{10} : \frac{243243 i ^{10}}{8192}

化简

\frac{16 !}{11 ! ( 16 - 11 ) !} (\frac{1}{2})^{5} (- \frac{3}{2} i)^{11} : - \frac{66339 3 i ^{11}}{4096}

化简

\frac{16 !}{12 ! ( 16 - 12 ) !} (\frac{1}{2})^{4} (- \frac{3}{2} i)^{12} : \frac{331695 i ^{12}}{16384}

化简

\frac{16 !}{13 ! ( 16 - 13 ) !} (\frac{1}{2})^{3} (- \frac{3}{2} i)^{13} : - \frac{25515 3 i ^{13}}{4096}

化简

\frac{16 !}{14 ! ( 16 - 14 ) !} (\frac{1}{2})^{2} (- \frac{3}{2} i)^{14} : \frac{32805 i ^{14}}{8192}

化简

\frac{16 !}{15 ! ( 16 - 15 ) !} (\frac{1}{2})^{1} (- \frac{3}{2} i)^{15} : - \frac{2187 3 i ^{15}}{4096}

化简

\frac{16 !}{16 ! ( 16 - 16 ) !} (\frac{1}{2})^{0} (- \frac{3}{2} i)^{16} : \frac{6561 i ^{16}}{65536}

= \frac{1}{65536} - \frac{3 i}{4096} + \frac{45 i ^{2}}{8192} - \frac{105 3 i ^{3}}{4096} + \frac{4095 i ^{4}}{16384} - \frac{2457 3 i ^{5}}{4096} + \frac{27027 i ^{6}}{8192} - \frac{19305 3 i ^{7}}{4096} + \frac{521235 i ^{8}}{32768} - \frac{57915 3 i ^{9}}{4096} + \frac{243243 i ^{10}}{8192} - \frac{66339 3 i ^{11}}{4096} + \frac{331695 i ^{12}}{16384} - \frac{25515 3 i ^{13}}{4096} + \frac{32805 i ^{14}}{8192} - \frac{2187 3 i ^{15}}{4096} + \frac{6561 i ^{16}}{65536}

化简

\frac{1}{65536} - \frac{3 i}{4096} + \frac{45 i ^{2}}{8192} - \frac{105 3 i ^{3}}{4096} + \frac{4095 i ^{4}}{16384} - \frac{2457 3 i ^{5}}{4096} + \frac{27027 i ^{6}}{8192} - \frac{19305 3 i ^{7}}{4096} + \frac{521235 i ^{8}}{32768} - \frac{57915 3 i ^{9}}{4096} + \frac{243243 i ^{10}}{8192} - \frac{66339 3 i ^{11}}{4096} + \frac{331695 i ^{12}}{16384} - \frac{25515 3 i ^{13}}{4096} + \frac{32805 i ^{14}}{8192} - \frac{2187 3 i ^{15}}{4096} + \frac{6561 i ^{16}}{65536} : - \frac{1}{2} + \frac{3}{2} i

= - \frac{1}{2} + \frac{3}{2} i

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