展開する (a^2-1/a)^4

解

展開する

(a^{2} - \frac{1}{a})^{4}

a^{8} - 4 a^{5} + 6 a^{2} - \frac{4}{a} + \frac{1}{a ^{4}}

解答ステップ

(a^{2} - \frac{1}{a})^{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 = a^{2}, b = - \frac{1}{a}

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

総和を展開する

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

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} (a^{2})^{4} (- \frac{1}{a})^{0} : a^{8}

簡素化

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

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (a^{2})^{2} (- \frac{1}{a})^{2} : 6 a^{2}

簡素化

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

簡素化

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

= a^{8} - 4 a^{5} + 6 a^{2} - \frac{4}{a} + \frac{1}{a ^{4}}