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受欢迎的代数 >

展开 (x^2+1/(x^2))^5

解答

展开

(x^{2} + \frac{1}{x ^{2}})^{5}

解答

x^{10} + 5 x^{6} + 10 x^{2} + \frac{10}{x ^{2}} + \frac{5}{x ^{6}} + \frac{1}{x ^{10}}

求解步骤

(x^{2} + \frac{1}{x ^{2}})^{5}

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 = \frac{1}{x ^{2}}

= i = 0 \sum 5 (i 5) (x^{2})^{(5 - i)} (\frac{1}{x ^{2}})^{i}

展开求和

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

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (x^{2})^{5} (\frac{1}{x ^{2}})^{0} : x^{10}

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} (x^{2})^{4} (\frac{1}{x ^{2}})^{1} : 5 x^{6}

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} (x^{2})^{3} (\frac{1}{x ^{2}})^{2} : 10 x^{2}

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} (x^{2})^{2} (\frac{1}{x ^{2}})^{3} : \frac{10}{x ^{2}}

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (x^{2})^{1} (\frac{1}{x ^{2}})^{4} : \frac{5}{x ^{6}}

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} (x^{2})^{0} (\frac{1}{x ^{2}})^{5} : \frac{1}{x ^{10}}

= x^{10} + 5 x^{6} + 10 x^{2} + \frac{10}{x ^{2}} + \frac{5}{x ^{6}} + \frac{1}{x ^{10}}

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