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受欢迎的代数 >

展开 (2x+k/x)^8

解答

展开

(2 x + \frac{k}{x})^{8}

解答

256 x^{8} + 1024 k x^{6} + 1792 k^{2} x^{4} + 1792 k^{3} x^{2} + 1120 k^{4} + \frac{448 k ^{5}}{x ^{2}} + \frac{112 k ^{6}}{x ^{4}} + \frac{16 k ^{7}}{x ^{6}} + \frac{k ^{8}}{x ^{8}}

求解步骤

(2 x + \frac{k}{x})^{8}

使用二项式定理:

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

a = 2 x, b = \frac{k}{x}

= i = 0 \sum 8 (i 8) (2 x)^{(8 - i)} (\frac{k}{x})^{i}

展开求和

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

化简

\frac{8 !}{0 ! ( 8 - 0 ) !} (2 x)^{8} (\frac{k}{x})^{0} : 256 x^{8}

化简

\frac{8 !}{1 ! ( 8 - 1 ) !} (2 x)^{7} (\frac{k}{x})^{1} : 1024 k x^{6}

化简

\frac{8 !}{2 ! ( 8 - 2 ) !} (2 x)^{6} (\frac{k}{x})^{2} : 1792 k^{2} x^{4}

化简

\frac{8 !}{3 ! ( 8 - 3 ) !} (2 x)^{5} (\frac{k}{x})^{3} : 1792 k^{3} x^{2}

化简

\frac{8 !}{4 ! ( 8 - 4 ) !} (2 x)^{4} (\frac{k}{x})^{4} : 1120 k^{4}

化简

\frac{8 !}{5 ! ( 8 - 5 ) !} (2 x)^{3} (\frac{k}{x})^{5} : \frac{448 k ^{5}}{x ^{2}}

化简

\frac{8 !}{6 ! ( 8 - 6 ) !} (2 x)^{2} (\frac{k}{x})^{6} : \frac{112 k ^{6}}{x ^{4}}

化简

\frac{8 !}{7 ! ( 8 - 7 ) !} (2 x)^{1} (\frac{k}{x})^{7} : \frac{16 k ^{7}}{x ^{6}}

化简

\frac{8 !}{8 ! ( 8 - 8 ) !} (2 x)^{0} (\frac{k}{x})^{8} : \frac{k ^{8}}{x ^{8}}

= 256 x^{8} + 1024 k x^{6} + 1792 k^{2} x^{4} + 1792 k^{3} x^{2} + 1120 k^{4} + \frac{448 k ^{5}}{x ^{2}} + \frac{112 k ^{6}}{x ^{4}} + \frac{16 k ^{7}}{x ^{6}} + \frac{k ^{8}}{x ^{8}}

流行的例子

展开 (1-e^{-s})/(s(s^2+3s+2))

e x p an d \frac{1 - e ^{- s}}{s ( s ^{2} + 3 s + 2 )}

展开 1/((x-5)(x^2+1))

e x p an d \frac{1}{( x - 5 ) ( x ^{2} + 1 )}

展开 (sqrt(1+2ax))/(\sqrt[3]{1+3bx)}

e x p an d \frac{1 + 2 a x}{3 1 + 3 b x}

展开 (pi+pie^{-s})/((s^2+4)(s^2+pi^2))

e x p an d \frac{π + π e ^{- s}}{( s ^{2} + 4 ) ( s ^{2} + π ^{2} )}

化简 (3-sqrt(x-1))^2

s im pl i f y (3 - x - 1)^{2}