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人気のある代数 >

展開する (sqrt(x)+(1/x))^8

解

展開する

(x + (\frac{1}{x}))^{8}

解

x^{4} + 8 x^{5} + 28 x + \frac{56}{x} + \frac{70}{x ^{2}} + \frac{56 x}{x ^{4}} + \frac{28}{x ^{5}} + \frac{8}{x ^{13}} + \frac{1}{x ^{8}}

解答ステップ

(x + (\frac{1}{x}))^{8}

簡素化

x + (\frac{1}{x}) : x + \frac{1}{x}

= (x + \frac{1}{x})^{8}

2項定理を適用する:

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

a = x, b = \frac{1}{x}

= i = 0 \sum 8 (i 8) (x)^{(8 - i)} (\frac{1}{x})^{i}

総和を展開する

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

簡素化

\frac{8 !}{0 ! ( 8 - 0 ) !} (x)^{8} (\frac{1}{x})^{0} : x^{4}

簡素化

\frac{8 !}{1 ! ( 8 - 1 ) !} (x)^{7} (\frac{1}{x})^{1} : \frac{8 ( x ) ^{7}}{x}

簡素化

\frac{8 !}{2 ! ( 8 - 2 ) !} (x)^{6} (\frac{1}{x})^{2} : 28 x

簡素化

\frac{8 !}{3 ! ( 8 - 3 ) !} (x)^{5} (\frac{1}{x})^{3} : \frac{56 ( x ) ^{5}}{x ^{3}}

簡素化

\frac{8 !}{4 ! ( 8 - 4 ) !} (x)^{4} (\frac{1}{x})^{4} : \frac{70}{x ^{2}}

簡素化

\frac{8 !}{5 ! ( 8 - 5 ) !} (x)^{3} (\frac{1}{x})^{5} : \frac{56 ( x ) ^{3}}{x ^{5}}

簡素化

\frac{8 !}{6 ! ( 8 - 6 ) !} (x)^{2} (\frac{1}{x})^{6} : \frac{28}{x ^{5}}

簡素化

\frac{8 !}{7 ! ( 8 - 7 ) !} (x)^{1} (\frac{1}{x})^{7} : \frac{8}{x ^{\frac{13}{2}}}

簡素化

\frac{8 !}{8 ! ( 8 - 8 ) !} (x)^{0} (\frac{1}{x})^{8} : \frac{1}{x ^{8}}

= x^{4} + \frac{8 ( x ) ^{7}}{x} + 28 x + \frac{56 ( x ) ^{5}}{x ^{3}} + \frac{70}{x ^{2}} + \frac{56 ( x ) ^{3}}{x ^{5}} + \frac{28}{x ^{5}} + \frac{8}{x ^{\frac{13}{2}}} + \frac{1}{x ^{8}}

簡素化

x^{4} + \frac{8 ( x ) ^{7}}{x} + 28 x + \frac{56 ( x ) ^{5}}{x ^{3}} + \frac{70}{x ^{2}} + \frac{56 ( x ) ^{3}}{x ^{5}} + \frac{28}{x ^{5}} + \frac{8}{x ^{\frac{13}{2}}} + \frac{1}{x ^{8}} : x^{4} + 8 x^{\frac{5}{2}} + 28 x + \frac{56}{x} + \frac{70}{x ^{2}} + \frac{56 x}{x ^{4}} + \frac{28}{x ^{5}} + \frac{8}{x ^{\frac{13}{2}}} + \frac{1}{x ^{8}}

= x^{4} + 8 x^{\frac{5}{2}} + 28 x + \frac{56}{x} + \frac{70}{x ^{2}} + \frac{56 x}{x ^{4}} + \frac{28}{x ^{5}} + \frac{8}{x ^{\frac{13}{2}}} + \frac{1}{x ^{8}}

簡素化

x^{4} + 8 x^{\frac{5}{2}} + 28 x + \frac{56}{x} + \frac{70}{x ^{2}} + \frac{56 x}{x ^{4}} + \frac{28}{x ^{5}} + \frac{8}{x ^{\frac{13}{2}}} + \frac{1}{x ^{8}} : x^{4} + 8 x^{5} + 28 x + \frac{56}{x} + \frac{70}{x ^{2}} + \frac{56 x}{x ^{4}} + \frac{28}{x ^{5}} + \frac{8}{x ^{13}} + \frac{1}{x ^{8}}

= x^{4} + 8 x^{5} + 28 x + \frac{56}{x} + \frac{70}{x ^{2}} + \frac{56 x}{x ^{4}} + \frac{28}{x ^{5}} + \frac{8}{x ^{13}} + \frac{1}{x ^{8}}

人気の例

簡約 (s+1)(s-1)(s+1/2)

s im pl i f y (s + 1) (s - 1) (s + \frac{1}{2})

展開する (2x-1/y)^9

e x p an d (2 x - \frac{1}{y})^{9}

展開する (50cos(8x))/((cos(x)-1)^2)

e x p an d \frac{50 cos ( 8 x )}{( cos ( x ) - 1 ) ^{2}}

展開する 2((-2x+2)(2x^2-1))

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

展開する sec(x)(1-sin(2x))

e x p an d sec (x) (1 - sin (2 x))