展開する (2x^2-3/x)^4

解

展開する

(2 x^{2} - \frac{3}{x})^{4}

16 x^{8} - 96 x^{5} + 216 x^{2} - \frac{216}{x} + \frac{81}{x ^{4}}

解答ステップ

(2 x^{2} - \frac{3}{x})^{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^{2}, b = - \frac{3}{x}

= i = 0 \sum 4 (i 4) (2 x^{2})^{(4 - i)} (- \frac{3}{x})^{i}

総和を展開する

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

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} (2 x^{2})^{4} (- \frac{3}{x})^{0} : 16 x^{8}

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} (2 x^{2})^{3} (- \frac{3}{x})^{1} : - 96 x^{5}

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (2 x^{2})^{2} (- \frac{3}{x})^{2} : 216 x^{2}

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} (2 x^{2})^{1} (- \frac{3}{x})^{3} : - \frac{216}{x}

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} (2 x^{2})^{0} (- \frac{3}{x})^{4} : \frac{81}{x ^{4}}

= 16 x^{8} - 96 x^{5} + 216 x^{2} - \frac{216}{x} + \frac{81}{x ^{4}}