展開する (1.6707100000000…+0.8976700000000…*i)^4

解

展開する

(1.6707100000000 \dots + 0.8976700000000 \dots \cdot i)^{4}

- 5.05490 \dots + 11.91074 \dots i

解答ステップ

(1.67071 \dots + 0.89767 \dots i)^{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 = 1.67071 \dots, b = 0.89767 \dots i

= i = 0 \sum 4 (i 4) \cdot 1.67071 \dots^{(4 - i)} (0.89767 \dots i)^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1.67071 \dots^{4} (0.89767 \dots i)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1.67071 \dots^{3} (0.89767 \dots i)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1.67071 \dots^{2} (0.89767 \dots i)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1.67071 \dots^{1} (0.89767 \dots i)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1.67071 \dots^{0} (0.89767 \dots i)^{4}

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1.67071 \dots^{4} (0.89767 \dots i)^{0} = 7.79119 \dots

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1.67071 \dots^{3} (0.89767 \dots i)^{1} : 16.74479 \dots i

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1.67071 \dots^{2} (0.89767 \dots i)^{2} = - 13.49543 \dots

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1.67071 \dots^{1} (0.89767 \dots i)^{3} : - 4.83405 \dots i

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1.67071 \dots^{0} (0.89767 \dots i)^{4} = 0.64933 \dots

= 7.79119 \dots + 16.74479 \dots i - 13.49543 \dots - 4.83405 \dots i + 0.64933 \dots

簡素化

7.79119 \dots + 16.74479 \dots i - 13.49543 \dots - 4.83405 \dots i + 0.64933 \dots : - 5.05490 \dots + 11.91074 \dots i

= - 5.05490 \dots + 11.91074 \dots i