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受欢迎的代数 >

展开 (3x-2/(9x^2))^9

解答

展开

(3 x - \frac{2}{9 x ^{2}})^{9}

解答

19683 x^{9} - 13122 x^{6} + 3888 x^{3} - 672 + \frac{224}{3 x ^{3}} - \frac{448}{81 x ^{6}} + \frac{1792}{6561 x ^{9}} - \frac{512}{59049 x ^{12}} + \frac{256}{1594323 x ^{15}} - \frac{512}{387420489 x ^{18}}

求解步骤

(3 x - \frac{2}{9 x ^{2}})^{9}

使用二项式定理:

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

a = 3 x, b = - \frac{2}{9 x ^{2}}

= i = 0 \sum 9 (i 9) (3 x)^{(9 - i)} (- \frac{2}{9 x ^{2}})^{i}

展开求和

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

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} (3 x)^{9} (- \frac{2}{9 x ^{2}})^{0} : 19683 x^{9}

化简

\frac{9 !}{1 ! ( 9 - 1 ) !} (3 x)^{8} (- \frac{2}{9 x ^{2}})^{1} : - 13122 x^{6}

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} (3 x)^{7} (- \frac{2}{9 x ^{2}})^{2} : 3888 x^{3}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} (3 x)^{6} (- \frac{2}{9 x ^{2}})^{3} : - 672

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} (3 x)^{5} (- \frac{2}{9 x ^{2}})^{4} : \frac{224}{3 x ^{3}}

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} (3 x)^{4} (- \frac{2}{9 x ^{2}})^{5} : - \frac{448}{81 x ^{6}}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} (3 x)^{3} (- \frac{2}{9 x ^{2}})^{6} : \frac{1792}{6561 x ^{9}}

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} (3 x)^{2} (- \frac{2}{9 x ^{2}})^{7} : - \frac{512}{59049 x ^{12}}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} (3 x)^{1} (- \frac{2}{9 x ^{2}})^{8} : \frac{256}{1594323 x ^{15}}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} (3 x)^{0} (- \frac{2}{9 x ^{2}})^{9} : - \frac{512}{387420489 x ^{18}}

= 19683 x^{9} - 13122 x^{6} + 3888 x^{3} - 672 + \frac{224}{3 x ^{3}} - \frac{448}{81 x ^{6}} + \frac{1792}{6561 x ^{9}} - \frac{512}{59049 x ^{12}} + \frac{256}{1594323 x ^{15}} - \frac{512}{387420489 x ^{18}}

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