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人気のある代数 >

展開する (2x^5+1/x)^{24}

解

展開する

(2 x^{5} + \frac{1}{x})^{24}

解

16777216 x^{120} + 201326592 x^{114} + 1157627904 x^{108} + 4244635648 x^{102} + 11142168576 x^{96} + 22284337152 x^{90} + 35283533824 x^{84} + 45364543488 x^{78} + 48199827456 x^{72} + 42844291072 x^{66} + 32133218304 x^{60} + 20448411648 x^{54} + 11076222976 x^{48} + 5112102912 x^{42} + 2008326144 x^{36} + 669442048 x^{30} + 188280576 x^{24} + 44301312 x^{18} + 8614144 x^{12} + 1360128 x^{6} + 170016 + \frac{16192}{x ^{6}} + \frac{1104}{x ^{12}} + \frac{48}{x ^{18}} + \frac{1}{x ^{24}}

解答ステップ

(2 x^{5} + \frac{1}{x})^{24}

2項定理を適用する:

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

a = 2 x^{5}, b = \frac{1}{x}

= i = 0 \sum 24 (i 24) (2 x^{5})^{(24 - i)} (\frac{1}{x})^{i}

総和を展開する

= \frac{24 !}{0 ! ( 24 - 0 ) !} (2 x^{5})^{24} (\frac{1}{x})^{0} + \frac{24 !}{1 ! ( 24 - 1 ) !} (2 x^{5})^{23} (\frac{1}{x})^{1} + \frac{24 !}{2 ! ( 24 - 2 ) !} (2 x^{5})^{22} (\frac{1}{x})^{2} + \frac{24 !}{3 ! ( 24 - 3 ) !} (2 x^{5})^{21} (\frac{1}{x})^{3} + \frac{24 !}{4 ! ( 24 - 4 ) !} (2 x^{5})^{20} (\frac{1}{x})^{4} + \frac{24 !}{5 ! ( 24 - 5 ) !} (2 x^{5})^{19} (\frac{1}{x})^{5} + \frac{24 !}{6 ! ( 24 - 6 ) !} (2 x^{5})^{18} (\frac{1}{x})^{6} + \frac{24 !}{7 ! ( 24 - 7 ) !} (2 x^{5})^{17} (\frac{1}{x})^{7} + \frac{24 !}{8 ! ( 24 - 8 ) !} (2 x^{5})^{16} (\frac{1}{x})^{8} + \frac{24 !}{9 ! ( 24 - 9 ) !} (2 x^{5})^{15} (\frac{1}{x})^{9} + \frac{24 !}{10 ! ( 24 - 10 ) !} (2 x^{5})^{14} (\frac{1}{x})^{10} + \frac{24 !}{11 ! ( 24 - 11 ) !} (2 x^{5})^{13} (\frac{1}{x})^{11} + \frac{24 !}{12 ! ( 24 - 12 ) !} (2 x^{5})^{12} (\frac{1}{x})^{12} + \frac{24 !}{13 ! ( 24 - 13 ) !} (2 x^{5})^{11} (\frac{1}{x})^{13} + \frac{24 !}{14 ! ( 24 - 14 ) !} (2 x^{5})^{10} (\frac{1}{x})^{14} + \frac{24 !}{15 ! ( 24 - 15 ) !} (2 x^{5})^{9} (\frac{1}{x})^{15} + \frac{24 !}{16 ! ( 24 - 16 ) !} (2 x^{5})^{8} (\frac{1}{x})^{16} + \frac{24 !}{17 ! ( 24 - 17 ) !} (2 x^{5})^{7} (\frac{1}{x})^{17} + \frac{24 !}{18 ! ( 24 - 18 ) !} (2 x^{5})^{6} (\frac{1}{x})^{18} + \frac{24 !}{19 ! ( 24 - 19 ) !} (2 x^{5})^{5} (\frac{1}{x})^{19} + \frac{24 !}{20 ! ( 24 - 20 ) !} (2 x^{5})^{4} (\frac{1}{x})^{20} + \frac{24 !}{21 ! ( 24 - 21 ) !} (2 x^{5})^{3} (\frac{1}{x})^{21} + \frac{24 !}{22 ! ( 24 - 22 ) !} (2 x^{5})^{2} (\frac{1}{x})^{22} + \frac{24 !}{23 ! ( 24 - 23 ) !} (2 x^{5})^{1} (\frac{1}{x})^{23} + \frac{24 !}{24 ! ( 24 - 24 ) !} (2 x^{5})^{0} (\frac{1}{x})^{24}

簡素化

\frac{24 !}{0 ! ( 24 - 0 ) !} (2 x^{5})^{24} (\frac{1}{x})^{0} : 16777216 x^{120}

簡素化

\frac{24 !}{1 ! ( 24 - 1 ) !} (2 x^{5})^{23} (\frac{1}{x})^{1} : 201326592 x^{114}

簡素化

\frac{24 !}{2 ! ( 24 - 2 ) !} (2 x^{5})^{22} (\frac{1}{x})^{2} : 1157627904 x^{108}

簡素化

\frac{24 !}{3 ! ( 24 - 3 ) !} (2 x^{5})^{21} (\frac{1}{x})^{3} : 4244635648 x^{102}

簡素化

\frac{24 !}{4 ! ( 24 - 4 ) !} (2 x^{5})^{20} (\frac{1}{x})^{4} : 11142168576 x^{96}

簡素化

\frac{24 !}{5 ! ( 24 - 5 ) !} (2 x^{5})^{19} (\frac{1}{x})^{5} : 22284337152 x^{90}

簡素化

\frac{24 !}{6 ! ( 24 - 6 ) !} (2 x^{5})^{18} (\frac{1}{x})^{6} : 35283533824 x^{84}

簡素化

\frac{24 !}{7 ! ( 24 - 7 ) !} (2 x^{5})^{17} (\frac{1}{x})^{7} : 45364543488 x^{78}

簡素化

\frac{24 !}{8 ! ( 24 - 8 ) !} (2 x^{5})^{16} (\frac{1}{x})^{8} : 48199827456 x^{72}

簡素化

\frac{24 !}{9 ! ( 24 - 9 ) !} (2 x^{5})^{15} (\frac{1}{x})^{9} : 42844291072 x^{66}

簡素化

\frac{24 !}{10 ! ( 24 - 10 ) !} (2 x^{5})^{14} (\frac{1}{x})^{10} : 32133218304 x^{60}

簡素化

\frac{24 !}{11 ! ( 24 - 11 ) !} (2 x^{5})^{13} (\frac{1}{x})^{11} : 20448411648 x^{54}

簡素化

\frac{24 !}{12 ! ( 24 - 12 ) !} (2 x^{5})^{12} (\frac{1}{x})^{12} : 11076222976 x^{48}

簡素化

\frac{24 !}{13 ! ( 24 - 13 ) !} (2 x^{5})^{11} (\frac{1}{x})^{13} : 5112102912 x^{42}

簡素化

\frac{24 !}{14 ! ( 24 - 14 ) !} (2 x^{5})^{10} (\frac{1}{x})^{14} : 2008326144 x^{36}

簡素化

\frac{24 !}{15 ! ( 24 - 15 ) !} (2 x^{5})^{9} (\frac{1}{x})^{15} : 669442048 x^{30}

簡素化

\frac{24 !}{16 ! ( 24 - 16 ) !} (2 x^{5})^{8} (\frac{1}{x})^{16} : 188280576 x^{24}

簡素化

\frac{24 !}{17 ! ( 24 - 17 ) !} (2 x^{5})^{7} (\frac{1}{x})^{17} : 44301312 x^{18}

簡素化

\frac{24 !}{18 ! ( 24 - 18 ) !} (2 x^{5})^{6} (\frac{1}{x})^{18} : 8614144 x^{12}

簡素化

\frac{24 !}{19 ! ( 24 - 19 ) !} (2 x^{5})^{5} (\frac{1}{x})^{19} : 1360128 x^{6}

簡素化

\frac{24 !}{20 ! ( 24 - 20 ) !} (2 x^{5})^{4} (\frac{1}{x})^{20} : 170016

簡素化

\frac{24 !}{21 ! ( 24 - 21 ) !} (2 x^{5})^{3} (\frac{1}{x})^{21} : \frac{16192}{x ^{6}}

簡素化

\frac{24 !}{22 ! ( 24 - 22 ) !} (2 x^{5})^{2} (\frac{1}{x})^{22} : \frac{1104}{x ^{12}}

簡素化

\frac{24 !}{23 ! ( 24 - 23 ) !} (2 x^{5})^{1} (\frac{1}{x})^{23} : \frac{48}{x ^{18}}

簡素化

\frac{24 !}{24 ! ( 24 - 24 ) !} (2 x^{5})^{0} (\frac{1}{x})^{24} : \frac{1}{x ^{24}}

= 16777216 x^{120} + 201326592 x^{114} + 1157627904 x^{108} + 4244635648 x^{102} + 11142168576 x^{96} + 22284337152 x^{90} + 35283533824 x^{84} + 45364543488 x^{78} + 48199827456 x^{72} + 42844291072 x^{66} + 32133218304 x^{60} + 20448411648 x^{54} + 11076222976 x^{48} + 5112102912 x^{42} + 2008326144 x^{36} + 669442048 x^{30} + 188280576 x^{24} + 44301312 x^{18} + 8614144 x^{12} + 1360128 x^{6} + 170016 + \frac{16192}{x ^{6}} + \frac{1104}{x ^{12}} + \frac{48}{x ^{18}} + \frac{1}{x ^{24}}

人気の例

展開する ((a+b+c)(-a+b+c)(a-b+c)(a+b-c))/(16)

e x p an d \frac{( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c )}{16}

展開する (-3+5)(-3+1)(-3-5)(-3-1)

e x p an d (- 3 + 5) (- 3 + 1) (- 3 - 5) (- 3 - 1)

簡約 (x-3)(x-1/5)

s im pl i f y (x - 3) (x - \frac{1}{5})

展開する (x+1)/((x+2)^2)

e x p an d \frac{x + 1}{( x + 2 ) ^{2}}

展開する (-x^2cos(x)+2(xsin(x)+cos(x))+C)/(x^2)

e x p an d \frac{- x ^{2} cos ( x ) + 2 ( x sin ( x ) + cos ( x ) ) + C}{x ^{2}}