Question

计算(1/i)*(根号2+根号2*i)^5+(1/1+i)^4+(1+i/1-i)^7的值

Answer 1

答案：
-1/4 - 16√2 + (16√2 - 1)i

过程：
(1/i) * (√2 + √2*i)^5 + [1/(1+i)]⁴ + [(1+i)/(1-i)]^7
= 1/√(-1) * 2^(5/2)*(1+i)^5 + [(1-i)/((1+i)(1-i))]⁴ + [(1+i)²/((1-i)(1+i))]^7
= √(-1)/[√(-1)√(-1)] * 4√2 * [-4(1+i)] -1/4 - i
= -i * 4√2 * [-4(1+i)] -1/4 - i
= -1/4 - 16√2 + (16√2 - 1)i

Notes：
(1+i)^5 = 1 + 5i + 10i² + 10i³ + 5i⁴ + i^5
= 1 + 5i + 10(-1) + 10(-1)i + 5(-1)(-1) + (-1)(-1)i
= 1 + 5i - 10 - 10i + 5 + i
= -4 - 4i
= -4(1+i)

[(1-i)/((1+i)(1-i))]⁴
= [(1-i)/(1-i²)]⁴
= [(1-i)/2]⁴
= (1/16)(1 - 4i + 6i² - 4i³ + i⁴)
= (1/16)(1 - 4i - 6 + 4i + 1)
= (1/16)(-4)
= -1/4

[(1+i)²/((1-i)(1+i))]^7
= [(1+i)²/2]^7
= (1/128)(1+i)^14
= (1/128)(-128i)
= -i