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受欢迎的代数 >

展开 (x^2+y^2)^4

解答

展开

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

解答

x^{8} + 4 x^{6} y^{2} + 6 x^{4} y^{4} + 4 x^{2} y^{6} + y^{8}

求解步骤

(x^{2} + y^{2})^{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 = y^{2}

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

展开求和

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

化简

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

化简

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

化简

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

化简

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

化简

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

= x^{8} + 4 x^{6} y^{2} + 6 x^{4} y^{4} + 4 x^{2} y^{6} + y^{8}

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