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受欢迎的代数 >

展开 (2x^6+(x^2)/3)^{10}

解答

展开

(2 x^{6} + \frac{x ^{2}}{3})^{10}

解答

1024 x^{60} + \frac{5120 x ^{56}}{3} + 1280 x^{52} + \frac{5120 x ^{48}}{9} + \frac{4480 x ^{44}}{27} + \frac{896 x ^{40}}{27} + \frac{1120 x ^{36}}{243} + \frac{320 x ^{32}}{729} + \frac{20 x ^{28}}{729} + \frac{20 x ^{24}}{19683} + \frac{x ^{20}}{59049}

求解步骤

(2 x^{6} + \frac{x ^{2}}{3})^{10}

使用二项式定理:

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

a = 2 x^{6}, b = \frac{x ^{2}}{3}

= i = 0 \sum 10 (i 10) (2 x^{6})^{(10 - i)} (\frac{x ^{2}}{3})^{i}

展开求和

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

化简

\frac{10 !}{0 ! ( 10 - 0 ) !} (2 x^{6})^{10} (\frac{x ^{2}}{3})^{0} : 1024 x^{60}

化简

\frac{10 !}{1 ! ( 10 - 1 ) !} (2 x^{6})^{9} (\frac{x ^{2}}{3})^{1} : \frac{5120 x ^{56}}{3}

化简

\frac{10 !}{2 ! ( 10 - 2 ) !} (2 x^{6})^{8} (\frac{x ^{2}}{3})^{2} : 1280 x^{52}

化简

\frac{10 !}{3 ! ( 10 - 3 ) !} (2 x^{6})^{7} (\frac{x ^{2}}{3})^{3} : \frac{5120 x ^{48}}{9}

化简

\frac{10 !}{4 ! ( 10 - 4 ) !} (2 x^{6})^{6} (\frac{x ^{2}}{3})^{4} : \frac{4480 x ^{44}}{27}

化简

\frac{10 !}{5 ! ( 10 - 5 ) !} (2 x^{6})^{5} (\frac{x ^{2}}{3})^{5} : \frac{896 x ^{40}}{27}

化简

\frac{10 !}{6 ! ( 10 - 6 ) !} (2 x^{6})^{4} (\frac{x ^{2}}{3})^{6} : \frac{1120 x ^{36}}{243}

化简

\frac{10 !}{7 ! ( 10 - 7 ) !} (2 x^{6})^{3} (\frac{x ^{2}}{3})^{7} : \frac{320 x ^{32}}{729}

化简

\frac{10 !}{8 ! ( 10 - 8 ) !} (2 x^{6})^{2} (\frac{x ^{2}}{3})^{8} : \frac{20 x ^{28}}{729}

化简

\frac{10 !}{9 ! ( 10 - 9 ) !} (2 x^{6})^{1} (\frac{x ^{2}}{3})^{9} : \frac{20 x ^{24}}{19683}

化简

\frac{10 !}{10 ! ( 10 - 10 ) !} (2 x^{6})^{0} (\frac{x ^{2}}{3})^{10} : \frac{x ^{20}}{59049}

= 1024 x^{60} + \frac{5120 x ^{56}}{3} + 1280 x^{52} + \frac{5120 x ^{48}}{9} + \frac{4480 x ^{44}}{27} + \frac{896 x ^{40}}{27} + \frac{1120 x ^{36}}{243} + \frac{320 x ^{32}}{729} + \frac{20 x ^{28}}{729} + \frac{20 x ^{24}}{19683} + \frac{x ^{20}}{59049}

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