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受欢迎的代数 >

展开 (1/(x^2)+y^3)^4

解答

展开

(\frac{1}{x ^{2}} + y^{3})^{4}

解答

\frac{1}{x ^{8}} + \frac{4 y ^{3}}{x ^{6}} + \frac{6 y ^{6}}{x ^{4}} + \frac{4 y ^{9}}{x ^{2}} + y^{12}

求解步骤

(\frac{1}{x ^{2}} + y^{3})^{4}

使用二项式定理:

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

a = \frac{1}{x ^{2}}, b = y^{3}

= i = 0 \sum 4 (i 4) (\frac{1}{x ^{2}})^{(4 - i)} (y^{3})^{i}

展开求和

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

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} (\frac{1}{x ^{2}})^{4} (y^{3})^{0} : \frac{1}{x ^{8}}

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} (\frac{1}{x ^{2}})^{3} (y^{3})^{1} : \frac{4 y ^{3}}{x ^{6}}

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} (\frac{1}{x ^{2}})^{2} (y^{3})^{2} : \frac{6 y ^{6}}{x ^{4}}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} (\frac{1}{x ^{2}})^{1} (y^{3})^{3} : \frac{4 y ^{9}}{x ^{2}}

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} (\frac{1}{x ^{2}})^{0} (y^{3})^{4} : y^{12}

= \frac{1}{x ^{8}} + \frac{4 y ^{3}}{x ^{6}} + \frac{6 y ^{6}}{x ^{4}} + \frac{4 y ^{9}}{x ^{2}} + y^{12}

流行的例子

展开 (1+2x^3)e^{x^2}

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展开 limx\Rightarrow 1(x^3+3x^2-x-3)(x^2-1)

e x p an d l im x \Rightarrow 1 (x^{3} + 3 x^{2} - x - 3) (x^{2} - 1)

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