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Popular Algebra >

expand (-1+2x^2)^{13}

Solution

expand

(- 1 + 2 x^{2})^{13}

Solution

- 1 + 26 x^{2} - 312 x^{4} + 2288 x^{6} - 11440 x^{8} + 41184 x^{10} - 109824 x^{12} + 219648 x^{14} - 329472 x^{16} + 366080 x^{18} - 292864 x^{20} + 159744 x^{22} - 53248 x^{24} + 8192 x^{26}

Solution steps

(- 1 + 2 x^{2})^{13}

Apply binomial theorem:

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

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

= i = 0 \sum 13 (i 13) (- 1)^{(13 - i)} (2 x^{2})^{i}

Expand summation

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

Simplify

\frac{13 !}{0 ! ( 13 - 0 ) !} (- 1)^{13} (2 x^{2})^{0} : - 1

Simplify

\frac{13 !}{1 ! ( 13 - 1 ) !} (- 1)^{12} (2 x^{2})^{1} : 26 x^{2}

Simplify

\frac{13 !}{2 ! ( 13 - 2 ) !} (- 1)^{11} (2 x^{2})^{2} : - 312 x^{4}

Simplify

\frac{13 !}{3 ! ( 13 - 3 ) !} (- 1)^{10} (2 x^{2})^{3} : 2288 x^{6}

Simplify

\frac{13 !}{4 ! ( 13 - 4 ) !} (- 1)^{9} (2 x^{2})^{4} : - 11440 x^{8}

Simplify

\frac{13 !}{5 ! ( 13 - 5 ) !} (- 1)^{8} (2 x^{2})^{5} : 41184 x^{10}

Simplify

\frac{13 !}{6 ! ( 13 - 6 ) !} (- 1)^{7} (2 x^{2})^{6} : - 109824 x^{12}

Simplify

\frac{13 !}{7 ! ( 13 - 7 ) !} (- 1)^{6} (2 x^{2})^{7} : 219648 x^{14}

Simplify

\frac{13 !}{8 ! ( 13 - 8 ) !} (- 1)^{5} (2 x^{2})^{8} : - 329472 x^{16}

Simplify

\frac{13 !}{9 ! ( 13 - 9 ) !} (- 1)^{4} (2 x^{2})^{9} : 366080 x^{18}

Simplify

\frac{13 !}{10 ! ( 13 - 10 ) !} (- 1)^{3} (2 x^{2})^{10} : - 292864 x^{20}

Simplify

\frac{13 !}{11 ! ( 13 - 11 ) !} (- 1)^{2} (2 x^{2})^{11} : 159744 x^{22}

Simplify

\frac{13 !}{12 ! ( 13 - 12 ) !} (- 1)^{1} (2 x^{2})^{12} : - 53248 x^{24}

Simplify

\frac{13 !}{13 ! ( 13 - 13 ) !} (- 1)^{0} (2 x^{2})^{13} : 8192 x^{26}

= - 1 + 26 x^{2} - 312 x^{4} + 2288 x^{6} - 11440 x^{8} + 41184 x^{10} - 109824 x^{12} + 219648 x^{14} - 329472 x^{16} + 366080 x^{18} - 292864 x^{20} + 159744 x^{22} - 53248 x^{24} + 8192 x^{26}

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