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受欢迎的代数 >

展开 (sqrt(x)+5/x)^9

解答

展开

(x + \frac{5}{x})^{9}

解答

x^{9} + 45 x^{3} + 900 x^{3} + 10500 + \frac{78750}{x ^{3}} + \frac{393750}{x ^{3}} + \frac{1312500 x}{x ^{5}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{15}} + \frac{1953125}{x ^{9}}

求解步骤

(x + \frac{5}{x})^{9}

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{5}{x}

= i = 0 \sum 9 (i 9) (x)^{(9 - i)} (\frac{5}{x})^{i}

展开求和

= \frac{9 !}{0 ! ( 9 - 0 ) !} (x)^{9} (\frac{5}{x})^{0} + \frac{9 !}{1 ! ( 9 - 1 ) !} (x)^{8} (\frac{5}{x})^{1} + \frac{9 !}{2 ! ( 9 - 2 ) !} (x)^{7} (\frac{5}{x})^{2} + \frac{9 !}{3 ! ( 9 - 3 ) !} (x)^{6} (\frac{5}{x})^{3} + \frac{9 !}{4 ! ( 9 - 4 ) !} (x)^{5} (\frac{5}{x})^{4} + \frac{9 !}{5 ! ( 9 - 5 ) !} (x)^{4} (\frac{5}{x})^{5} + \frac{9 !}{6 ! ( 9 - 6 ) !} (x)^{3} (\frac{5}{x})^{6} + \frac{9 !}{7 ! ( 9 - 7 ) !} (x)^{2} (\frac{5}{x})^{7} + \frac{9 !}{8 ! ( 9 - 8 ) !} (x)^{1} (\frac{5}{x})^{8} + \frac{9 !}{9 ! ( 9 - 9 ) !} (x)^{0} (\frac{5}{x})^{9}

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} (x)^{9} (\frac{5}{x})^{0} : (x)^{9}

化简

\frac{9 !}{1 ! ( 9 - 1 ) !} (x)^{8} (\frac{5}{x})^{1} : 45 x^{3}

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} (x)^{7} (\frac{5}{x})^{2} : \frac{900 ( x ) ^{7}}{x ^{2}}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} (x)^{6} (\frac{5}{x})^{3} : 10500

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} (x)^{5} (\frac{5}{x})^{4} : \frac{78750 ( x ) ^{5}}{x ^{4}}

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} (x)^{4} (\frac{5}{x})^{5} : \frac{393750}{x ^{3}}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} (x)^{3} (\frac{5}{x})^{6} : \frac{1312500 ( x ) ^{3}}{x ^{6}}

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} (x)^{2} (\frac{5}{x})^{7} : \frac{2812500}{x ^{6}}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} (x)^{1} (\frac{5}{x})^{8} : \frac{3515625}{x ^{\frac{15}{2}}}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} (x)^{0} (\frac{5}{x})^{9} : \frac{1953125}{x ^{9}}

= (x)^{9} + 45 x^{3} + \frac{900 ( x ) ^{7}}{x ^{2}} + 10500 + \frac{78750 ( x ) ^{5}}{x ^{4}} + \frac{393750}{x ^{3}} + \frac{1312500 ( x ) ^{3}}{x ^{6}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{\frac{15}{2}}} + \frac{1953125}{x ^{9}}

化简

(x)^{9} + 45 x^{3} + \frac{900 ( x ) ^{7}}{x ^{2}} + 10500 + \frac{78750 ( x ) ^{5}}{x ^{4}} + \frac{393750}{x ^{3}} + \frac{1312500 ( x ) ^{3}}{x ^{6}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{\frac{15}{2}}} + \frac{1953125}{x ^{9}} : x^{\frac{9}{2}} + 45 x^{3} + 900 x^{\frac{3}{2}} + 10500 + \frac{78750}{x ^{\frac{3}{2}}} + \frac{393750}{x ^{3}} + \frac{1312500 x}{x ^{5}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{\frac{15}{2}}} + \frac{1953125}{x ^{9}}

= x^{\frac{9}{2}} + 45 x^{3} + 900 x^{\frac{3}{2}} + 10500 + \frac{78750}{x ^{\frac{3}{2}}} + \frac{393750}{x ^{3}} + \frac{1312500 x}{x ^{5}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{\frac{15}{2}}} + \frac{1953125}{x ^{9}}

化简

x^{\frac{9}{2}} + 45 x^{3} + 900 x^{\frac{3}{2}} + 10500 + \frac{78750}{x ^{\frac{3}{2}}} + \frac{393750}{x ^{3}} + \frac{1312500 x}{x ^{5}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{\frac{15}{2}}} + \frac{1953125}{x ^{9}} : x^{9} + 45 x^{3} + 900 x^{3} + 10500 + \frac{78750}{x ^{3}} + \frac{393750}{x ^{3}} + \frac{1312500 x}{x ^{5}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{15}} + \frac{1953125}{x ^{9}}

= x^{9} + 45 x^{3} + 900 x^{3} + 10500 + \frac{78750}{x ^{3}} + \frac{393750}{x ^{3}} + \frac{1312500 x}{x ^{5}} + \frac{2812500}{x ^{6}} + \frac{3515625}{x ^{15}} + \frac{1953125}{x ^{9}}

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