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受欢迎的代数 >

展开 (2+x/2)^6

解答

展开

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

解答

64 + 96 x + 60 x^{2} + 20 x^{3} + \frac{15 x ^{4}}{4} + \frac{3 x ^{5}}{8} + \frac{x ^{6}}{64}

求解步骤

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

使用二项式定理:

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

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

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

展开求和

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

化简

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

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 2^{5} (\frac{x}{2})^{1} : 96 x

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 2^{4} (\frac{x}{2})^{2} : 60 x^{2}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 2^{3} (\frac{x}{2})^{3} : 20 x^{3}

化简

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

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 2^{1} (\frac{x}{2})^{5} : \frac{3 x ^{5}}{8}

化简

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

= 64 + 96 x + 60 x^{2} + 20 x^{3} + \frac{15 x ^{4}}{4} + \frac{3 x ^{5}}{8} + \frac{x ^{6}}{64}

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