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受欢迎的代数 >

展开 (y^{3/4}-2x^{3/4})^4

解答

展开

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

解答

y^{3} - 8 y^{\frac{9}{4}} x^{\frac{3}{4}} + 24 y^{\frac{3}{2}} x^{\frac{3}{2}} - 32 y^{\frac{3}{4}} x^{\frac{9}{4}} + 16 x^{3}

求解步骤

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

使用二项式定理:

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

a = y^{\frac{3}{4}}, b = - 2 x^{\frac{3}{4}}

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

展开求和

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

化简

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

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} (y^{\frac{3}{4}})^{3} (- 2 x^{\frac{3}{4}})^{1} : - 8 y^{\frac{9}{4}} x^{\frac{3}{4}}

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} (y^{\frac{3}{4}})^{2} (- 2 x^{\frac{3}{4}})^{2} : 24 y^{\frac{3}{2}} x^{\frac{3}{2}}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} (y^{\frac{3}{4}})^{1} (- 2 x^{\frac{3}{4}})^{3} : - 32 y^{\frac{3}{4}} x^{\frac{9}{4}}

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} (y^{\frac{3}{4}})^{0} (- 2 x^{\frac{3}{4}})^{4} : 16 x^{3}

= y^{3} - 8 y^{\frac{9}{4}} x^{\frac{3}{4}} + 24 y^{\frac{3}{2}} x^{\frac{3}{2}} - 32 y^{\frac{3}{4}} x^{\frac{9}{4}} + 16 x^{3}

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