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受欢迎的代数 >

展开 (3/4 e^t+1/4)^{10}

解答

展开

(\frac{3}{4} e^{t} + \frac{1}{4})^{10}

解答

\frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t}}{524288} + \frac{76545 e ^{6 t}}{524288} + \frac{8505 e ^{4 t}}{524288} + \frac{15 e ^{t}}{524288} + \frac{295245 e ^{8 t}}{1048576} + \frac{405 e ^{2 t}}{1048576} + \frac{1}{1048576} + \frac{32805 e ^{7 t}}{131072} + \frac{405 e ^{3 t}}{131072} + \frac{15309 e ^{5 t}}{262144}

求解步骤

(\frac{3}{4} e^{t} + \frac{1}{4})^{10}

使用二项式定理:

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

a = \frac{3}{4} e^{t}, b = \frac{1}{4}

= i = 0 \sum 10 (i 10) (\frac{3}{4} e^{t})^{(10 - i)} (\frac{1}{4})^{i}

展开求和

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

化简

\frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{3}{4} e^{t})^{10} (\frac{1}{4})^{0} : \frac{59049}{1048576} e^{10 t}

化简

\frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{3}{4} e^{t})^{9} (\frac{1}{4})^{1} : \frac{98415 e ^{9 t}}{524288}

化简

\frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{3}{4} e^{t})^{8} (\frac{1}{4})^{2} : \frac{295245 e ^{8 t}}{1048576}

化简

\frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{3}{4} e^{t})^{7} (\frac{1}{4})^{3} : \frac{32805 e ^{7 t}}{131072}

化简

\frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{3}{4} e^{t})^{6} (\frac{1}{4})^{4} : \frac{76545 e ^{6 t}}{524288}

化简

\frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{3}{4} e^{t})^{5} (\frac{1}{4})^{5} : \frac{15309 e ^{5 t}}{262144}

化简

\frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{3}{4} e^{t})^{4} (\frac{1}{4})^{6} : \frac{8505 e ^{4 t}}{524288}

化简

\frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{3}{4} e^{t})^{3} (\frac{1}{4})^{7} : \frac{405 e ^{3 t}}{131072}

化简

\frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{3}{4} e^{t})^{2} (\frac{1}{4})^{8} : \frac{405 e ^{2 t}}{1048576}

化简

\frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{3}{4} e^{t})^{1} (\frac{1}{4})^{9} : \frac{15 e ^{t}}{524288}

\frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{3}{4} e^{t})^{0} (\frac{1}{4})^{10} = \frac{1}{1048576}

= \frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t}}{524288} + \frac{295245 e ^{8 t}}{1048576} + \frac{32805 e ^{7 t}}{131072} + \frac{76545 e ^{6 t}}{524288} + \frac{15309 e ^{5 t}}{262144} + \frac{8505 e ^{4 t}}{524288} + \frac{405 e ^{3 t}}{131072} + \frac{405 e ^{2 t}}{1048576} + \frac{15 e ^{t}}{524288} + \frac{1}{1048576}

化简

\frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t}}{524288} + \frac{295245 e ^{8 t}}{1048576} + \frac{32805 e ^{7 t}}{131072} + \frac{76545 e ^{6 t}}{524288} + \frac{15309 e ^{5 t}}{262144} + \frac{8505 e ^{4 t}}{524288} + \frac{405 e ^{3 t}}{131072} + \frac{405 e ^{2 t}}{1048576} + \frac{15 e ^{t}}{524288} + \frac{1}{1048576} : \frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t} + 76545 e ^{6 t} + 8505 e ^{4 t} + 15 e ^{t}}{524288} + \frac{295245 e ^{8 t} + 405 e ^{2 t} + 1}{1048576} + \frac{32805 e ^{7 t} + 405 e ^{3 t}}{131072} + \frac{15309 e ^{5 t}}{262144}

= \frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t} + 76545 e ^{6 t} + 8505 e ^{4 t} + 15 e ^{t}}{524288} + \frac{295245 e ^{8 t} + 405 e ^{2 t} + 1}{1048576} + \frac{32805 e ^{7 t} + 405 e ^{3 t}}{131072} + \frac{15309 e ^{5 t}}{262144}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{98415 e ^{9 t} + 76545 e ^{6 t} + 8505 e ^{4 t} + 15 e ^{t}}{524288} = \frac{98415 e ^{9 t}}{524288} + \frac{76545 e ^{6 t}}{524288} + \frac{8505 e ^{4 t}}{524288} + \frac{15 e ^{t}}{524288}

= \frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t}}{524288} + \frac{76545 e ^{6 t}}{524288} + \frac{8505 e ^{4 t}}{524288} + \frac{15 e ^{t}}{524288} + \frac{295245 e ^{8 t} + 405 e ^{2 t} + 1}{1048576} + \frac{32805 e ^{7 t} + 405 e ^{3 t}}{131072} + \frac{15309 e ^{5 t}}{262144}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{295245 e ^{8 t} + 405 e ^{2 t} + 1}{1048576} = \frac{295245 e ^{8 t}}{1048576} + \frac{405 e ^{2 t}}{1048576} + \frac{1}{1048576}

= \frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t}}{524288} + \frac{76545 e ^{6 t}}{524288} + \frac{8505 e ^{4 t}}{524288} + \frac{15 e ^{t}}{524288} + \frac{295245 e ^{8 t}}{1048576} + \frac{405 e ^{2 t}}{1048576} + \frac{1}{1048576} + \frac{32805 e ^{7 t} + 405 e ^{3 t}}{131072} + \frac{15309 e ^{5 t}}{262144}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{32805 e ^{7 t} + 405 e ^{3 t}}{131072} = \frac{32805 e ^{7 t}}{131072} + \frac{405 e ^{3 t}}{131072}

= \frac{59049}{1048576} e^{10 t} + \frac{98415 e ^{9 t}}{524288} + \frac{76545 e ^{6 t}}{524288} + \frac{8505 e ^{4 t}}{524288} + \frac{15 e ^{t}}{524288} + \frac{295245 e ^{8 t}}{1048576} + \frac{405 e ^{2 t}}{1048576} + \frac{1}{1048576} + \frac{32805 e ^{7 t}}{131072} + \frac{405 e ^{3 t}}{131072} + \frac{15309 e ^{5 t}}{262144}

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