展開する (a^2-1/4)^4

解

展開する

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

a^{8} - a^{6} + \frac{3 a ^{4}}{8} - \frac{a ^{2}}{16} + \frac{1}{256}

解答ステップ

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

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

総和を展開する

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

簡素化

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

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} (a^{2})^{3} (- \frac{1}{4})^{1} : - a^{6}

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (a^{2})^{2} (- \frac{1}{4})^{2} : \frac{3 a ^{4}}{8}

簡素化

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

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} (a^{2})^{0} (- \frac{1}{4})^{4} : \frac{1}{256}

= a^{8} - a^{6} + \frac{3 a ^{4}}{8} - \frac{a ^{2}}{16} + \frac{1}{256}