What is expand (1/(x^3)-2x)^{16} ?

The solution to expand (1/(x^3)-2x)^{16} is

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expand (1/(x^3)-2x)^{16}

expand

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

\frac{1}{x ^{48}} - \frac{32}{x ^{44}} + \frac{480}{x ^{40}} - \frac{4480}{x ^{36}} + \frac{29120}{x ^{32}} - \frac{139776}{x ^{28}} + \frac{512512}{x ^{24}} - \frac{1464320}{x ^{20}} + \frac{3294720}{x ^{16}} - \frac{5857280}{x ^{12}} + \frac{8200192}{x ^{8}} - \frac{8945664}{x ^{4}} + 7454720 - 4587520 x^{4} + 1966080 x^{8} - 524288 x^{12} + 65536 x^{16}

Solution steps

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

Apply binomial theorem:

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

a = \frac{1}{x ^{3}}, b = - 2 x

= i = 0 \sum 16 (i 16) (\frac{1}{x ^{3}})^{(16 - i)} (- 2 x)^{i}

Expand summation

= \frac{16 !}{0 ! ( 16 - 0 ) !} (\frac{1}{x ^{3}})^{16} (- 2 x)^{0} + \frac{16 !}{1 ! ( 16 - 1 ) !} (\frac{1}{x ^{3}})^{15} (- 2 x)^{1} + \frac{16 !}{2 ! ( 16 - 2 ) !} (\frac{1}{x ^{3}})^{14} (- 2 x)^{2} + \frac{16 !}{3 ! ( 16 - 3 ) !} (\frac{1}{x ^{3}})^{13} (- 2 x)^{3} + \frac{16 !}{4 ! ( 16 - 4 ) !} (\frac{1}{x ^{3}})^{12} (- 2 x)^{4} + \frac{16 !}{5 ! ( 16 - 5 ) !} (\frac{1}{x ^{3}})^{11} (- 2 x)^{5} + \frac{16 !}{6 ! ( 16 - 6 ) !} (\frac{1}{x ^{3}})^{10} (- 2 x)^{6} + \frac{16 !}{7 ! ( 16 - 7 ) !} (\frac{1}{x ^{3}})^{9} (- 2 x)^{7} + \frac{16 !}{8 ! ( 16 - 8 ) !} (\frac{1}{x ^{3}})^{8} (- 2 x)^{8} + \frac{16 !}{9 ! ( 16 - 9 ) !} (\frac{1}{x ^{3}})^{7} (- 2 x)^{9} + \frac{16 !}{10 ! ( 16 - 10 ) !} (\frac{1}{x ^{3}})^{6} (- 2 x)^{10} + \frac{16 !}{11 ! ( 16 - 11 ) !} (\frac{1}{x ^{3}})^{5} (- 2 x)^{11} + \frac{16 !}{12 ! ( 16 - 12 ) !} (\frac{1}{x ^{3}})^{4} (- 2 x)^{12} + \frac{16 !}{13 ! ( 16 - 13 ) !} (\frac{1}{x ^{3}})^{3} (- 2 x)^{13} + \frac{16 !}{14 ! ( 16 - 14 ) !} (\frac{1}{x ^{3}})^{2} (- 2 x)^{14} + \frac{16 !}{15 ! ( 16 - 15 ) !} (\frac{1}{x ^{3}})^{1} (- 2 x)^{15} + \frac{16 !}{16 ! ( 16 - 16 ) !} (\frac{1}{x ^{3}})^{0} (- 2 x)^{16}

Simplify

\frac{16 !}{0 ! ( 16 - 0 ) !} (\frac{1}{x ^{3}})^{16} (- 2 x)^{0} : \frac{1}{x ^{48}}

Simplify

\frac{16 !}{1 ! ( 16 - 1 ) !} (\frac{1}{x ^{3}})^{15} (- 2 x)^{1} : - \frac{32}{x ^{44}}

Simplify

\frac{16 !}{2 ! ( 16 - 2 ) !} (\frac{1}{x ^{3}})^{14} (- 2 x)^{2} : \frac{480}{x ^{40}}

Simplify

\frac{16 !}{3 ! ( 16 - 3 ) !} (\frac{1}{x ^{3}})^{13} (- 2 x)^{3} : - \frac{4480}{x ^{36}}

Simplify

\frac{16 !}{4 ! ( 16 - 4 ) !} (\frac{1}{x ^{3}})^{12} (- 2 x)^{4} : \frac{29120}{x ^{32}}

Simplify

\frac{16 !}{5 ! ( 16 - 5 ) !} (\frac{1}{x ^{3}})^{11} (- 2 x)^{5} : - \frac{139776}{x ^{28}}

Simplify

\frac{16 !}{6 ! ( 16 - 6 ) !} (\frac{1}{x ^{3}})^{10} (- 2 x)^{6} : \frac{512512}{x ^{24}}

Simplify

\frac{16 !}{7 ! ( 16 - 7 ) !} (\frac{1}{x ^{3}})^{9} (- 2 x)^{7} : - \frac{1464320}{x ^{20}}

Simplify

\frac{16 !}{8 ! ( 16 - 8 ) !} (\frac{1}{x ^{3}})^{8} (- 2 x)^{8} : \frac{3294720}{x ^{16}}

Simplify

\frac{16 !}{9 ! ( 16 - 9 ) !} (\frac{1}{x ^{3}})^{7} (- 2 x)^{9} : - \frac{5857280}{x ^{12}}

Simplify

\frac{16 !}{10 ! ( 16 - 10 ) !} (\frac{1}{x ^{3}})^{6} (- 2 x)^{10} : \frac{8200192}{x ^{8}}

Simplify

\frac{16 !}{11 ! ( 16 - 11 ) !} (\frac{1}{x ^{3}})^{5} (- 2 x)^{11} : - \frac{8945664}{x ^{4}}

Simplify

\frac{16 !}{12 ! ( 16 - 12 ) !} (\frac{1}{x ^{3}})^{4} (- 2 x)^{12} : 7454720

Simplify

\frac{16 !}{13 ! ( 16 - 13 ) !} (\frac{1}{x ^{3}})^{3} (- 2 x)^{13} : - 4587520 x^{4}

Simplify

\frac{16 !}{14 ! ( 16 - 14 ) !} (\frac{1}{x ^{3}})^{2} (- 2 x)^{14} : 1966080 x^{8}

Simplify

\frac{16 !}{15 ! ( 16 - 15 ) !} (\frac{1}{x ^{3}})^{1} (- 2 x)^{15} : - 524288 x^{12}

Simplify

\frac{16 !}{16 ! ( 16 - 16 ) !} (\frac{1}{x ^{3}})^{0} (- 2 x)^{16} : 65536 x^{16}

= \frac{1}{x ^{48}} - \frac{32}{x ^{44}} + \frac{480}{x ^{40}} - \frac{4480}{x ^{36}} + \frac{29120}{x ^{32}} - \frac{139776}{x ^{28}} + \frac{512512}{x ^{24}} - \frac{1464320}{x ^{20}} + \frac{3294720}{x ^{16}} - \frac{5857280}{x ^{12}} + \frac{8200192}{x ^{8}} - \frac{8945664}{x ^{4}} + 7454720 - 4587520 x^{4} + 1966080 x^{8} - 524288 x^{12} + 65536 x^{16}

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