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受欢迎的代数 >

展开 (1-1/x)^4

解答

展开

(1 - \frac{1}{x})^{4}

解答

1 - \frac{4}{x} + \frac{6}{x ^{2}} - \frac{4}{x ^{3}} + \frac{1}{x ^{4}}

求解步骤

(1 - \frac{1}{x})^{4}

使用二项式定理:

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

a = 1, b = - \frac{1}{x}

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

展开求和

= \frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} (- \frac{1}{x})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (- \frac{1}{x})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1^{2} (- \frac{1}{x})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (- \frac{1}{x})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} (- \frac{1}{x})^{4}

化简

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

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (- \frac{1}{x})^{1} : - \frac{4}{x}

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1^{2} (- \frac{1}{x})^{2} : \frac{6}{x ^{2}}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (- \frac{1}{x})^{3} : - \frac{4}{x ^{3}}

化简

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

= 1 - \frac{4}{x} + \frac{6}{x ^{2}} - \frac{4}{x ^{3}} + \frac{1}{x ^{4}}

流行的例子

展开 ((r_{1}r_{2}c+L)(1-w^2Lc))/(c^2w)

e x p an d \frac{( r _{1} r _{2} c + L ) ( 1 - w ^{2} L c )}{c ^{2} w}

展开 ((a+2)!+a!+(a+1)!)/((a+2)*a!)

e x p an d \frac{( a + 2 ) ! + a ! + ( a + 1 ) !}{( a + 2 ) \cdot a !}

展开 (2s^3-26s^2+102s-124)/((s-3)^3(s-4))

e x p an d \frac{2 s ^{3} - 26 s ^{2} + 102 s - 124}{( s - 3 ) ^{3} ( s - 4 )}

展开 (\partial)/(\partial y)(5e^y)+e^y(5x+5y)

e x p an d \frac{\partial}{\partial y} (5 e^{y}) + e^{y} (5 x + 5 y)

展开 ((2x^2))/((1-x)^2)

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