展開する (1+4/(x^2))^4

解

展開する

(1 + \frac{4}{x ^{2}})^{4}

1 + \frac{16}{x ^{2}} + \frac{96}{x ^{4}} + \frac{256}{x ^{6}} + \frac{256}{x ^{8}}

解答ステップ

(1 + \frac{4}{x ^{2}})^{4}

2項定理を適用する:

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

a = 1, b = \frac{4}{x ^{2}}

= i = 0 \sum 4 (i 4) \cdot 1^{(4 - i)} (\frac{4}{x ^{2}})^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} (\frac{4}{x ^{2}})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (\frac{4}{x ^{2}})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1^{2} (\frac{4}{x ^{2}})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (\frac{4}{x ^{2}})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} (\frac{4}{x ^{2}})^{4}

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} (\frac{4}{x ^{2}})^{0} : 1

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (\frac{4}{x ^{2}})^{1} : \frac{16}{x ^{2}}

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1^{2} (\frac{4}{x ^{2}})^{2} : \frac{96}{x ^{4}}

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (\frac{4}{x ^{2}})^{3} : \frac{256}{x ^{6}}

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} (\frac{4}{x ^{2}})^{4} : \frac{256}{x ^{8}}

= 1 + \frac{16}{x ^{2}} + \frac{96}{x ^{4}} + \frac{256}{x ^{6}} + \frac{256}{x ^{8}}