What is expand (2x-5/(x^2))^{12} ?

The solution to expand (2x-5/(x^2))^{12} is

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expand (2x-5/(x^2))^{12}

expand

(2 x - \frac{5}{x ^{2}})^{12}

4096 x^{12} - 122880 x^{9} + 1689600 x^{6} - 14080000 x^{3} + 79200000 - \frac{316800000}{x ^{3}} + \frac{924000000}{x ^{6}} - \frac{1980000000}{x ^{9}} + \frac{3093750000}{x ^{12}} - \frac{3437500000}{x ^{15}} + \frac{2578125000}{x ^{18}} - \frac{1171875000}{x ^{21}} + \frac{244140625}{x ^{24}}

Solution steps

(2 x - \frac{5}{x ^{2}})^{12}

Apply binomial theorem:

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

a = 2 x, b = - \frac{5}{x ^{2}}

= i = 0 \sum 12 (i 12) (2 x)^{(12 - i)} (- \frac{5}{x ^{2}})^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} (2 x)^{12} (- \frac{5}{x ^{2}})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (2 x)^{11} (- \frac{5}{x ^{2}})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (2 x)^{10} (- \frac{5}{x ^{2}})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (2 x)^{9} (- \frac{5}{x ^{2}})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (2 x)^{8} (- \frac{5}{x ^{2}})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (2 x)^{7} (- \frac{5}{x ^{2}})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (2 x)^{6} (- \frac{5}{x ^{2}})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (2 x)^{5} (- \frac{5}{x ^{2}})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (2 x)^{4} (- \frac{5}{x ^{2}})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (2 x)^{3} (- \frac{5}{x ^{2}})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (2 x)^{2} (- \frac{5}{x ^{2}})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (2 x)^{1} (- \frac{5}{x ^{2}})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (2 x)^{0} (- \frac{5}{x ^{2}})^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x)^{12} (- \frac{5}{x ^{2}})^{0} : 4096 x^{12}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x)^{11} (- \frac{5}{x ^{2}})^{1} : - 122880 x^{9}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x)^{10} (- \frac{5}{x ^{2}})^{2} : 1689600 x^{6}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x)^{9} (- \frac{5}{x ^{2}})^{3} : - 14080000 x^{3}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x)^{8} (- \frac{5}{x ^{2}})^{4} : 79200000

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x)^{7} (- \frac{5}{x ^{2}})^{5} : - \frac{316800000}{x ^{3}}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x)^{6} (- \frac{5}{x ^{2}})^{6} : \frac{924000000}{x ^{6}}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x)^{5} (- \frac{5}{x ^{2}})^{7} : - \frac{1980000000}{x ^{9}}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x)^{4} (- \frac{5}{x ^{2}})^{8} : \frac{3093750000}{x ^{12}}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x)^{3} (- \frac{5}{x ^{2}})^{9} : - \frac{3437500000}{x ^{15}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x)^{2} (- \frac{5}{x ^{2}})^{10} : \frac{2578125000}{x ^{18}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x)^{1} (- \frac{5}{x ^{2}})^{11} : - \frac{1171875000}{x ^{21}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (2 x)^{0} (- \frac{5}{x ^{2}})^{12} : \frac{244140625}{x ^{24}}

= 4096 x^{12} - 122880 x^{9} + 1689600 x^{6} - 14080000 x^{3} + 79200000 - \frac{316800000}{x ^{3}} + \frac{924000000}{x ^{6}} - \frac{1980000000}{x ^{9}} + \frac{3093750000}{x ^{12}} - \frac{3437500000}{x ^{15}} + \frac{2578125000}{x ^{18}} - \frac{1171875000}{x ^{21}} + \frac{244140625}{x ^{24}}

\frac{\partial}{\partial x} ((z) (x + y) e^{y})

(1 + 5)^{4}

y (2 x - y + 1) d x - x (3 x - 4 y + 3) d y

6 (2 - 2 - 6 - 3)

y (2 e^{2 x} y = \frac{x e ^{2 x}}{2} - \frac{e ^{2 x}}{4} + c)