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受欢迎的代数 >

展开 (1/(2sqrt(x))-1/2)^5

解答

展开

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

解答

\frac{1}{32 x ^{2} x} - \frac{5}{32 x ^{2}} + \frac{5}{16 x x} - \frac{5}{16 x} + \frac{5}{32 x} - \frac{1}{32}

求解步骤

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

使用二项式定理:

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

a = \frac{1}{2 x}, b = - \frac{1}{2}

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

展开求和

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

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{2 x})^{5} (- \frac{1}{2})^{0} : \frac{1}{32 ( x ) ^{5}}

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{2 x})^{4} (- \frac{1}{2})^{1} : - \frac{5}{32 x ^{2}}

化简

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

化简

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

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (\frac{1}{2 x})^{1} (- \frac{1}{2})^{4} : \frac{5}{32 x}

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} (\frac{1}{2 x})^{0} (- \frac{1}{2})^{5} : - \frac{1}{32}

= \frac{1}{32 ( x ) ^{5}} - \frac{5}{32 x ^{2}} + \frac{5}{16 ( x ) ^{3}} - \frac{5}{16 x} + \frac{5}{32 x} - \frac{1}{32}

化简

\frac{1}{32 ( x ) ^{5}} - \frac{5}{32 x ^{2}} + \frac{5}{16 ( x ) ^{3}} - \frac{5}{16 x} + \frac{5}{32 x} - \frac{1}{32} : \frac{1}{32 x ^{2} x} - \frac{5}{32 x ^{2}} + \frac{5}{16 x x} - \frac{5}{16 x} + \frac{5}{32 x} - \frac{1}{32}

= \frac{1}{32 x ^{2} x} - \frac{5}{32 x ^{2}} + \frac{5}{16 x x} - \frac{5}{16 x} + \frac{5}{32 x} - \frac{1}{32}

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