展開する (2x+(15)/(x^2))^4

解

展開する

(2 x + \frac{15}{x ^{2}})^{4}

16 x^{4} + 480 x + \frac{5400}{x ^{2}} + \frac{27000}{x ^{5}} + \frac{50625}{x ^{8}}

解答ステップ

(2 x + \frac{15}{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 = 2 x, b = \frac{15}{x ^{2}}

= i = 0 \sum 4 (i 4) (2 x)^{(4 - i)} (\frac{15}{x ^{2}})^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (\frac{15}{x ^{2}})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (\frac{15}{x ^{2}})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (\frac{15}{x ^{2}})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (\frac{15}{x ^{2}})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (2 x)^{0} (\frac{15}{x ^{2}})^{4}

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (\frac{15}{x ^{2}})^{0} : 16 x^{4}

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (\frac{15}{x ^{2}})^{1} : 480 x

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (\frac{15}{x ^{2}})^{2} : \frac{5400}{x ^{2}}

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (\frac{15}{x ^{2}})^{3} : \frac{27000}{x ^{5}}

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} (2 x)^{0} (\frac{15}{x ^{2}})^{4} : \frac{50625}{x ^{8}}

= 16 x^{4} + 480 x + \frac{5400}{x ^{2}} + \frac{27000}{x ^{5}} + \frac{50625}{x ^{8}}