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Popular >

expand (2x^2-1/x)^{12}

Solution

expand

(2 x^{2} - \frac{1}{x})^{12}

Solution

4096 x^{24} - 24576 x^{21} + 67584 x^{18} - 112640 x^{15} + 126720 x^{12} - 101376 x^{9} + 59136 x^{6} - 25344 x^{3} + 7920 - \frac{1760}{x ^{3}} + \frac{264}{x ^{6}} - \frac{24}{x ^{9}} + \frac{1}{x ^{12}}

Solution steps

(2 x^{2} - \frac{1}{x})^{12}

Apply binomial theorem:

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

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

= i = 0 \sum 12 (i 12) (2 x^{2})^{(12 - i)} (- \frac{1}{x})^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x^{2})^{12} (- \frac{1}{x})^{0} : 4096 x^{24}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x^{2})^{11} (- \frac{1}{x})^{1} : - 24576 x^{21}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x^{2})^{10} (- \frac{1}{x})^{2} : 67584 x^{18}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x^{2})^{9} (- \frac{1}{x})^{3} : - 112640 x^{15}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x^{2})^{8} (- \frac{1}{x})^{4} : 126720 x^{12}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x^{2})^{7} (- \frac{1}{x})^{5} : - 101376 x^{9}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x^{2})^{6} (- \frac{1}{x})^{6} : 59136 x^{6}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x^{2})^{5} (- \frac{1}{x})^{7} : - 25344 x^{3}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x^{2})^{4} (- \frac{1}{x})^{8} : 7920

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x^{2})^{3} (- \frac{1}{x})^{9} : - \frac{1760}{x ^{3}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x^{2})^{2} (- \frac{1}{x})^{10} : \frac{264}{x ^{6}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x^{2})^{1} (- \frac{1}{x})^{11} : - \frac{24}{x ^{9}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (2 x^{2})^{0} (- \frac{1}{x})^{12} : \frac{1}{x ^{12}}

= 4096 x^{24} - 24576 x^{21} + 67584 x^{18} - 112640 x^{15} + 126720 x^{12} - 101376 x^{9} + 59136 x^{6} - 25344 x^{3} + 7920 - \frac{1760}{x ^{3}} + \frac{264}{x ^{6}} - \frac{24}{x ^{9}} + \frac{1}{x ^{12}}

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