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受欢迎的代数 >

展开 (3/c+(c^2)/4)^7

解答

展开

(\frac{3}{c} + \frac{c ^{2}}{4})^{7}

解答

\frac{2187}{c ^{7}} + \frac{5103}{4 c ^{4}} + \frac{5103}{16 c} + \frac{2835 c ^{2}}{64} + \frac{945 c ^{5}}{256} + \frac{189 c ^{8}}{1024} + \frac{21 c ^{11}}{4096} + \frac{c ^{14}}{16384}

求解步骤

(\frac{3}{c} + \frac{c ^{2}}{4})^{7}

使用二项式定理:

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

a = \frac{3}{c}, b = \frac{c ^{2}}{4}

= i = 0 \sum 7 (i 7) (\frac{3}{c})^{(7 - i)} (\frac{c ^{2}}{4})^{i}

展开求和

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

化简

\frac{7 !}{0 ! ( 7 - 0 ) !} (\frac{3}{c})^{7} (\frac{c ^{2}}{4})^{0} : \frac{2187}{c ^{7}}

化简

\frac{7 !}{1 ! ( 7 - 1 ) !} (\frac{3}{c})^{6} (\frac{c ^{2}}{4})^{1} : \frac{5103}{4 c ^{4}}

化简

\frac{7 !}{2 ! ( 7 - 2 ) !} (\frac{3}{c})^{5} (\frac{c ^{2}}{4})^{2} : \frac{5103}{16 c}

化简

\frac{7 !}{3 ! ( 7 - 3 ) !} (\frac{3}{c})^{4} (\frac{c ^{2}}{4})^{3} : \frac{2835 c ^{2}}{64}

化简

\frac{7 !}{4 ! ( 7 - 4 ) !} (\frac{3}{c})^{3} (\frac{c ^{2}}{4})^{4} : \frac{945 c ^{5}}{256}

化简

\frac{7 !}{5 ! ( 7 - 5 ) !} (\frac{3}{c})^{2} (\frac{c ^{2}}{4})^{5} : \frac{189 c ^{8}}{1024}

化简

\frac{7 !}{6 ! ( 7 - 6 ) !} (\frac{3}{c})^{1} (\frac{c ^{2}}{4})^{6} : \frac{21 c ^{11}}{4096}

化简

\frac{7 !}{7 ! ( 7 - 7 ) !} (\frac{3}{c})^{0} (\frac{c ^{2}}{4})^{7} : \frac{c ^{14}}{16384}

= \frac{2187}{c ^{7}} + \frac{5103}{4 c ^{4}} + \frac{5103}{16 c} + \frac{2835 c ^{2}}{64} + \frac{945 c ^{5}}{256} + \frac{189 c ^{8}}{1024} + \frac{21 c ^{11}}{4096} + \frac{c ^{14}}{16384}

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