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受欢迎的代数 >

展开 (2a^3+3/(sqrt(a)))^6

解答

展开

(2 a^{3} + \frac{3}{a})^{6}

解答

64 a^{18} + 576 a^{29} + 2160 a^{11} + 4320 a^{15} + 2916 a + 4860 a^{4} + \frac{729}{a ^{3}}

求解步骤

(2 a^{3} + \frac{3}{a})^{6}

使用二项式定理:

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

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

= i = 0 \sum 6 (i 6) (2 a^{3})^{(6 - i)} (\frac{3}{a})^{i}

展开求和

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

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} (2 a^{3})^{6} (\frac{3}{a})^{0} : 64 a^{18}

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} (2 a^{3})^{5} (\frac{3}{a})^{1} : 576 a^{\frac{29}{2}}

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} (2 a^{3})^{4} (\frac{3}{a})^{2} : 2160 a^{11}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} (2 a^{3})^{3} (\frac{3}{a})^{3} : \frac{4320 a ^{9}}{( a ) ^{3}}

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} (2 a^{3})^{2} (\frac{3}{a})^{4} : 4860 a^{4}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} (2 a^{3})^{1} (\frac{3}{a})^{5} : \frac{2916 a ^{3}}{( a ) ^{5}}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} (2 a^{3})^{0} (\frac{3}{a})^{6} : \frac{729}{a ^{3}}

= 64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + \frac{4320 a ^{9}}{( a ) ^{3}} + 4860 a^{4} + \frac{2916 a ^{3}}{( a ) ^{5}} + \frac{729}{a ^{3}}

\frac{4320 a ^{9}}{( a ) ^{3}} = \frac{4320 a ^{8}}{a}

\frac{2916 a ^{3}}{( a ) ^{5}} = \frac{2916 a}{a}

= 64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + \frac{4320 a ^{8}}{a} + 4860 a^{4} + \frac{2916 a}{a} + \frac{729}{a ^{3}}

合并

64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + \frac{4320 a ^{8}}{a} + 4860 a^{4} + \frac{2916 a}{a} + \frac{729}{a ^{3}} : 64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + 108 a (40 a^{7} + 27) + 4860 a^{4} + \frac{729}{a ^{3}}

= 64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + 108 a (40 a^{7} + 27) + 4860 a^{4} + \frac{729}{a ^{3}}

展开

108 a (40 a^{7} + 27) : 4320 a^{\frac{15}{2}} + 2916 a

= 64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + 4320 a^{\frac{15}{2}} + 2916 a + 4860 a^{4} + \frac{729}{a ^{3}}

化简

64 a^{18} + 576 a^{\frac{29}{2}} + 2160 a^{11} + 4320 a^{\frac{15}{2}} + 2916 a + 4860 a^{4} + \frac{729}{a ^{3}} : 64 a^{18} + 576 a^{29} + 2160 a^{11} + 4320 a^{15} + 2916 a + 4860 a^{4} + \frac{729}{a ^{3}}

= 64 a^{18} + 576 a^{29} + 2160 a^{11} + 4320 a^{15} + 2916 a + 4860 a^{4} + \frac{729}{a ^{3}}

流行的例子

展开 x^2dx+y(x-1)dy

e x p an d x^{2} d x + y (x - 1) d y

展开 ln(\sqrt[3]{x})

e x p an d ln (3 x)

展开 (1+ln(x))^2

e x p an d (1 + ln (x))^{2}

展开 ((n+1)(n+2)(2n+3))/6

e x p an d \frac{( n + 1 ) ( n + 2 ) ( 2 n + 3 )}{6}

化简 (4+2i)^4

s im pl i f y (4 + 2 i)^{4}