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受欢迎的代数 >

展开 ((x^2)/2+a/x)^6

解答

展开

(\frac{x ^{2}}{2} + \frac{a}{x})^{6}

解答

\frac{x ^{12}}{64} + \frac{3 a x ^{9} + 15 a ^{2} x ^{6}}{16} + \frac{5 a ^{3} x ^{3}}{2} + \frac{15 a ^{4}}{4} + \frac{3 a ^{5}}{x ^{3}} + \frac{a ^{6}}{x ^{6}}

求解步骤

(\frac{x ^{2}}{2} + \frac{a}{x})^{6}

使用二项式定理:

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

a = \frac{x ^{2}}{2}, b = \frac{a}{x}

= i = 0 \sum 6 (i 6) (\frac{x ^{2}}{2})^{(6 - i)} (\frac{a}{x})^{i}

展开求和

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

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{x ^{2}}{2})^{6} (\frac{a}{x})^{0} : \frac{x ^{12}}{64}

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{x ^{2}}{2})^{5} (\frac{a}{x})^{1} : \frac{3 a x ^{9}}{16}

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{x ^{2}}{2})^{4} (\frac{a}{x})^{2} : \frac{15 a ^{2} x ^{6}}{16}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{x ^{2}}{2})^{3} (\frac{a}{x})^{3} : \frac{5 a ^{3} x ^{3}}{2}

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{x ^{2}}{2})^{2} (\frac{a}{x})^{4} : \frac{15 a ^{4}}{4}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{x ^{2}}{2})^{1} (\frac{a}{x})^{5} : \frac{3 a ^{5}}{x ^{3}}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{x ^{2}}{2})^{0} (\frac{a}{x})^{6} : \frac{a ^{6}}{x ^{6}}

= \frac{x ^{12}}{64} + \frac{3 a x ^{9}}{16} + \frac{15 a ^{2} x ^{6}}{16} + \frac{5 a ^{3} x ^{3}}{2} + \frac{15 a ^{4}}{4} + \frac{3 a ^{5}}{x ^{3}} + \frac{a ^{6}}{x ^{6}}

合并

\frac{x ^{12}}{64} + \frac{3 a x ^{9}}{16} + \frac{15 a ^{2} x ^{6}}{16} + \frac{5 a ^{3} x ^{3}}{2} + \frac{15 a ^{4}}{4} + \frac{3 a ^{5}}{x ^{3}} + \frac{a ^{6}}{x ^{6}} : \frac{x ^{12}}{64} + \frac{3 a x ^{9} + 15 a ^{2} x ^{6}}{16} + \frac{5 a ^{3} x ^{3}}{2} + \frac{15 a ^{4}}{4} + \frac{3 a ^{5}}{x ^{3}} + \frac{a ^{6}}{x ^{6}}

= \frac{x ^{12}}{64} + \frac{3 a x ^{9} + 15 a ^{2} x ^{6}}{16} + \frac{5 a ^{3} x ^{3}}{2} + \frac{15 a ^{4}}{4} + \frac{3 a ^{5}}{x ^{3}} + \frac{a ^{6}}{x ^{6}}

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