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受欢迎的代数 >

展开 (x+1/y)^7

解答

展开

(x + \frac{1}{y})^{7}

解答

x^{7} + \frac{7 x ^{6}}{y} + \frac{21 x ^{5}}{y ^{2}} + \frac{35 x ^{4}}{y ^{3}} + \frac{35 x ^{3}}{y ^{4}} + \frac{21 x ^{2}}{y ^{5}} + \frac{7 x}{y ^{6}} + \frac{1}{y ^{7}}

求解步骤

(x + \frac{1}{y})^{7}

使用二项式定理:

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

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

= i = 0 \sum 7 (i 7) x^{(7 - i)} (\frac{1}{y})^{i}

展开求和

= \frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (\frac{1}{y})^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (\frac{1}{y})^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (\frac{1}{y})^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (\frac{1}{y})^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (\frac{1}{y})^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (\frac{1}{y})^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (\frac{1}{y})^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (\frac{1}{y})^{7}

化简

\frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (\frac{1}{y})^{0} : x^{7}

化简

\frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (\frac{1}{y})^{1} : \frac{7 x ^{6}}{y}

化简

\frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (\frac{1}{y})^{2} : \frac{21 x ^{5}}{y ^{2}}

化简

\frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (\frac{1}{y})^{3} : \frac{35 x ^{4}}{y ^{3}}

化简

\frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (\frac{1}{y})^{4} : \frac{35 x ^{3}}{y ^{4}}

化简

\frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (\frac{1}{y})^{5} : \frac{21 x ^{2}}{y ^{5}}

化简

\frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (\frac{1}{y})^{6} : \frac{7 x}{y ^{6}}

化简

\frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (\frac{1}{y})^{7} : \frac{1}{y ^{7}}

= x^{7} + \frac{7 x ^{6}}{y} + \frac{21 x ^{5}}{y ^{2}} + \frac{35 x ^{4}}{y ^{3}} + \frac{35 x ^{3}}{y ^{4}} + \frac{21 x ^{2}}{y ^{5}} + \frac{7 x}{y ^{6}} + \frac{1}{y ^{7}}

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