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受欢迎的代数 >

展开 (3x+(5/6)/x)^8

解答

展开

(3 x + \frac{\frac{5}{6}}{x})^{8}

解答

6561 x^{8} + 14580 x^{6} + 14175 x^{4} + 7875 x^{2} + \frac{21875}{8} + \frac{21875}{36 x ^{2}} + \frac{109375}{1296 x ^{4}} + \frac{78125}{11664 x ^{6}} + \frac{390625}{1679616 x ^{8}}

求解步骤

(3 x + \frac{\frac{5}{6}}{x})^{8}

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{\frac{5}{6}}{x}

= i = 0 \sum 8 (i 8) (3 x)^{(8 - i)} (\frac{\frac{5}{6}}{x})^{i}

展开求和

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

化简

\frac{8 !}{0 ! ( 8 - 0 ) !} (3 x)^{8} (\frac{\frac{5}{6}}{x})^{0} : 6561 x^{8}

化简

\frac{8 !}{1 ! ( 8 - 1 ) !} (3 x)^{7} (\frac{\frac{5}{6}}{x})^{1} : 14580 x^{6}

化简

\frac{8 !}{2 ! ( 8 - 2 ) !} (3 x)^{6} (\frac{\frac{5}{6}}{x})^{2} : 14175 x^{4}

化简

\frac{8 !}{3 ! ( 8 - 3 ) !} (3 x)^{5} (\frac{\frac{5}{6}}{x})^{3} : 7875 x^{2}

化简

\frac{8 !}{4 ! ( 8 - 4 ) !} (3 x)^{4} (\frac{\frac{5}{6}}{x})^{4} : \frac{21875}{8}

化简

\frac{8 !}{5 ! ( 8 - 5 ) !} (3 x)^{3} (\frac{\frac{5}{6}}{x})^{5} : \frac{21875}{36 x ^{2}}

化简

\frac{8 !}{6 ! ( 8 - 6 ) !} (3 x)^{2} (\frac{\frac{5}{6}}{x})^{6} : \frac{109375}{1296 x ^{4}}

化简

\frac{8 !}{7 ! ( 8 - 7 ) !} (3 x)^{1} (\frac{\frac{5}{6}}{x})^{7} : \frac{78125}{11664 x ^{6}}

化简

\frac{8 !}{8 ! ( 8 - 8 ) !} (3 x)^{0} (\frac{\frac{5}{6}}{x})^{8} : \frac{390625}{1679616 x ^{8}}

= 6561 x^{8} + 14580 x^{6} + 14175 x^{4} + 7875 x^{2} + \frac{21875}{8} + \frac{21875}{36 x ^{2}} + \frac{109375}{1296 x ^{4}} + \frac{78125}{11664 x ^{6}} + \frac{390625}{1679616 x ^{8}}

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