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受欢迎的代数 >

展开 (4x+1/(kx))^7

解答

展开

(4 x + \frac{1}{k x})^{7}

解答

16384 x^{7} + \frac{28672 x ^{5}}{k} + \frac{21504 x ^{3}}{k ^{2}} + \frac{8960 x}{k ^{3}} + \frac{2240}{k ^{4} x} + \frac{336}{k ^{5} x ^{3}} + \frac{28}{k ^{6} x ^{5}} + \frac{1}{k ^{7} x ^{7}}

求解步骤

(4 x + \frac{1}{k x})^{7}

使用二项式定理:

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

a = 4 x, b = \frac{1}{k x}

= i = 0 \sum 7 (i 7) (4 x)^{(7 - i)} (\frac{1}{k x})^{i}

展开求和

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

化简

\frac{7 !}{0 ! ( 7 - 0 ) !} (4 x)^{7} (\frac{1}{k x})^{0} : 16384 x^{7}

化简

\frac{7 !}{1 ! ( 7 - 1 ) !} (4 x)^{6} (\frac{1}{k x})^{1} : \frac{28672 x ^{5}}{k}

化简

\frac{7 !}{2 ! ( 7 - 2 ) !} (4 x)^{5} (\frac{1}{k x})^{2} : \frac{21504 x ^{3}}{k ^{2}}

化简

\frac{7 !}{3 ! ( 7 - 3 ) !} (4 x)^{4} (\frac{1}{k x})^{3} : \frac{8960 x}{k ^{3}}

化简

\frac{7 !}{4 ! ( 7 - 4 ) !} (4 x)^{3} (\frac{1}{k x})^{4} : \frac{2240}{k ^{4} x}

化简

\frac{7 !}{5 ! ( 7 - 5 ) !} (4 x)^{2} (\frac{1}{k x})^{5} : \frac{336}{k ^{5} x ^{3}}

化简

\frac{7 !}{6 ! ( 7 - 6 ) !} (4 x)^{1} (\frac{1}{k x})^{6} : \frac{28}{k ^{6} x ^{5}}

化简

\frac{7 !}{7 ! ( 7 - 7 ) !} (4 x)^{0} (\frac{1}{k x})^{7} : \frac{1}{k ^{7} x ^{7}}

= 16384 x^{7} + \frac{28672 x ^{5}}{k} + \frac{21504 x ^{3}}{k ^{2}} + \frac{8960 x}{k ^{3}} + \frac{2240}{k ^{4} x} + \frac{336}{k ^{5} x ^{3}} + \frac{28}{k ^{6} x ^{5}} + \frac{1}{k ^{7} x ^{7}}

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