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

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  • 答案:

    -1/4 - 16√2 + (16√2 - 1)i

    (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