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受欢迎的微积分 >

展开 (x^2+x^3)^4

解答

展开

(x^{2} + x^{3})^{4}

解答

x^{8} + 4 x^{9} + 6 x^{10} + 4 x^{11} + x^{12}

求解步骤

(x^{2} + x^{3})^{4}

使用二项式定理:

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

a = x^{2}, b = x^{3}

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

展开求和

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

化简

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

化简

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

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} (x^{2})^{2} (x^{3})^{2} : 6 x^{10}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} (x^{2})^{1} (x^{3})^{3} : 4 x^{11}

化简

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

= x^{8} + 4 x^{9} + 6 x^{10} + 4 x^{11} + x^{12}

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