展开 ((5x^2+3x^3)^7)/7

解答

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\frac{( 5 x ^{2} + 3 x ^{3} ) ^{7}}{7}

\frac{2187 x ^{21}}{7} + 3645 x^{20} + 18225 x^{19} + 50625 x^{18} + 84375 x^{17} + 84375 x^{16} + 46875 x^{15} + \frac{78125 x ^{14}}{7}

求解步骤

\frac{( 5 x ^{2} + 3 x ^{3} ) ^{7}}{7}

(5 x^{2} + 3 x^{3})^{7} = 78125 x^{14} + 328125 x^{15} + 590625 x^{16} + 590625 x^{17} + 354375 x^{18} + 127575 x^{19} + 25515 x^{20} + 2187 x^{21}

= \frac{78125 x ^{14} + 328125 x ^{15} + 590625 x ^{16} + 590625 x ^{17} + 354375 x ^{18} + 127575 x ^{19} + 25515 x ^{20} + 2187 x ^{21}}{7}

使用分式法则:

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

\frac{78125 x ^{14} + 328125 x ^{15} + 590625 x ^{16} + 590625 x ^{17} + 354375 x ^{18} + 127575 x ^{19} + 25515 x ^{20} + 2187 x ^{21}}{7} = \frac{78125 x ^{14}}{7} + \frac{328125 x ^{15}}{7} + \frac{590625 x ^{16}}{7} + \frac{590625 x ^{17}}{7} + \frac{354375 x ^{18}}{7} + \frac{127575 x ^{19}}{7} + \frac{25515 x ^{20}}{7} + \frac{2187 x ^{21}}{7}

= \frac{78125 x ^{14}}{7} + \frac{328125 x ^{15}}{7} + \frac{590625 x ^{16}}{7} + \frac{590625 x ^{17}}{7} + \frac{354375 x ^{18}}{7} + \frac{127575 x ^{19}}{7} + \frac{25515 x ^{20}}{7} + \frac{2187 x ^{21}}{7}

\frac{328125 x ^{15}}{7} = 46875 x^{15}

\frac{590625 x ^{16}}{7} = 84375 x^{16}

\frac{590625 x ^{17}}{7} = 84375 x^{17}

\frac{354375 x ^{18}}{7} = 50625 x^{18}

\frac{127575 x ^{19}}{7} = 18225 x^{19}

\frac{25515 x ^{20}}{7} = 3645 x^{20}

= \frac{78125 x ^{14}}{7} + 46875 x^{15} + 84375 x^{16} + 84375 x^{17} + 50625 x^{18} + 18225 x^{19} + 3645 x^{20} + \frac{2187 x ^{21}}{7}

改写为标准形式

= \frac{2187 x ^{21}}{7} + 3645 x^{20} + 18225 x^{19} + 50625 x^{18} + 84375 x^{17} + 84375 x^{16} + 46875 x^{15} + \frac{78125 x ^{14}}{7}

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