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受欢迎的代数 >

展开 (x^{1/3}+y^{1/3})^6

解答

展开

(x^{\frac{1}{3}} + y^{\frac{1}{3}})^{6}

解答

x^{2} + 6 x^{\frac{5}{3}} y^{\frac{1}{3}} + 15 x^{\frac{4}{3}} y^{\frac{2}{3}} + 20 x y + 15 x^{\frac{2}{3}} y^{\frac{4}{3}} + 6 x^{\frac{1}{3}} y^{\frac{5}{3}} + y^{2}

求解步骤

(x^{\frac{1}{3}} + y^{\frac{1}{3}})^{6}

使用二项式定理:

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

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

= i = 0 \sum 6 (i 6) (x^{\frac{1}{3}})^{(6 - i)} (y^{\frac{1}{3}})^{i}

展开求和

= \frac{6 !}{0 ! ( 6 - 0 ) !} (x^{\frac{1}{3}})^{6} (y^{\frac{1}{3}})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (x^{\frac{1}{3}})^{5} (y^{\frac{1}{3}})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (x^{\frac{1}{3}})^{4} (y^{\frac{1}{3}})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (x^{\frac{1}{3}})^{3} (y^{\frac{1}{3}})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (x^{\frac{1}{3}})^{2} (y^{\frac{1}{3}})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (x^{\frac{1}{3}})^{1} (y^{\frac{1}{3}})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (x^{\frac{1}{3}})^{0} (y^{\frac{1}{3}})^{6}

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} (x^{\frac{1}{3}})^{6} (y^{\frac{1}{3}})^{0} : x^{2}

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} (x^{\frac{1}{3}})^{5} (y^{\frac{1}{3}})^{1} : 6 x^{\frac{5}{3}} y^{\frac{1}{3}}

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} (x^{\frac{1}{3}})^{4} (y^{\frac{1}{3}})^{2} : 15 x^{\frac{4}{3}} y^{\frac{2}{3}}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} (x^{\frac{1}{3}})^{3} (y^{\frac{1}{3}})^{3} : 20 x y

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} (x^{\frac{1}{3}})^{2} (y^{\frac{1}{3}})^{4} : 15 x^{\frac{2}{3}} y^{\frac{4}{3}}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} (x^{\frac{1}{3}})^{1} (y^{\frac{1}{3}})^{5} : 6 x^{\frac{1}{3}} y^{\frac{5}{3}}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} (x^{\frac{1}{3}})^{0} (y^{\frac{1}{3}})^{6} : y^{2}

= x^{2} + 6 x^{\frac{5}{3}} y^{\frac{1}{3}} + 15 x^{\frac{4}{3}} y^{\frac{2}{3}} + 20 x y + 15 x^{\frac{2}{3}} y^{\frac{4}{3}} + 6 x^{\frac{1}{3}} y^{\frac{5}{3}} + y^{2}

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