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Popular >

expand (1/2 n-2v)^{10}

Solution

expand

(\frac{1}{2} n - 2 v)^{10}

Solution

\frac{n ^{10}}{1024} - \frac{5 n ^{9} v}{128} + \frac{45 n ^{8} v ^{2}}{64} + \frac{- 15 n ^{7} v ^{3} + 105 n ^{6} v ^{4}}{2} - 252 n^{5} v^{5} + 840 n^{4} v^{6} - 1920 n^{3} v^{7} + 2880 n^{2} v^{8} - 2560 n v^{9} + 1024 v^{10}

Solution steps

(\frac{1}{2} n - 2 v)^{10}

Apply binomial theorem:

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

a = \frac{1}{2} n, b = - 2 v

= i = 0 \sum 10 (i 10) (\frac{1}{2} n)^{(10 - i)} (- 2 v)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{1}{2} n)^{10} (- 2 v)^{0} : \frac{n ^{10}}{1024}

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{1}{2} n)^{9} (- 2 v)^{1} : - \frac{5 n ^{9} v}{128}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{1}{2} n)^{8} (- 2 v)^{2} : \frac{45 n ^{8} v ^{2}}{64}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{1}{2} n)^{7} (- 2 v)^{3} : - \frac{15 n ^{7} v ^{3}}{2}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{1}{2} n)^{6} (- 2 v)^{4} : \frac{105 n ^{6} v ^{4}}{2}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{1}{2} n)^{5} (- 2 v)^{5} : - 252 n^{5} v^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{1}{2} n)^{4} (- 2 v)^{6} : 840 n^{4} v^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{1}{2} n)^{3} (- 2 v)^{7} : - 1920 n^{3} v^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{1}{2} n)^{2} (- 2 v)^{8} : 2880 n^{2} v^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{1}{2} n)^{1} (- 2 v)^{9} : - 2560 n v^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{1}{2} n)^{0} (- 2 v)^{10} : 1024 v^{10}

= \frac{n ^{10}}{1024} - \frac{5 n ^{9} v}{128} + \frac{45 n ^{8} v ^{2}}{64} - \frac{15 n ^{7} v ^{3}}{2} + \frac{105 n ^{6} v ^{4}}{2} - 252 n^{5} v^{5} + 840 n^{4} v^{6} - 1920 n^{3} v^{7} + 2880 n^{2} v^{8} - 2560 n v^{9} + 1024 v^{10}

Join

\frac{n ^{10}}{1024} - \frac{5 n ^{9} v}{128} + \frac{45 n ^{8} v ^{2}}{64} - \frac{15 n ^{7} v ^{3}}{2} + \frac{105 n ^{6} v ^{4}}{2} - 252 n^{5} v^{5} + 840 n^{4} v^{6} - 1920 n^{3} v^{7} + 2880 n^{2} v^{8} - 2560 n v^{9} + 1024 v^{10} : \frac{n ^{10}}{1024} - \frac{5 n ^{9} v}{128} + \frac{45 n ^{8} v ^{2}}{64} + \frac{- 15 n ^{7} v ^{3} + 105 n ^{6} v ^{4}}{2} - 252 n^{5} v^{5} + 840 n^{4} v^{6} - 1920 n^{3} v^{7} + 2880 n^{2} v^{8} - 2560 n v^{9} + 1024 v^{10}

= \frac{n ^{10}}{1024} - \frac{5 n ^{9} v}{128} + \frac{45 n ^{8} v ^{2}}{64} + \frac{- 15 n ^{7} v ^{3} + 105 n ^{6} v ^{4}}{2} - 252 n^{5} v^{5} + 840 n^{4} v^{6} - 1920 n^{3} v^{7} + 2880 n^{2} v^{8} - 2560 n v^{9} + 1024 v^{10}

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