 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013-resit/1/solution

Squaring first is simpler than expanding the tenth powers. Since

(1 - i)^{2} = - 2 i

and

(1 + i)^{2} = 2 i

, while

i^{4} = 1

, the requested Cartesian forms are

((1 - i)^{2})^{5} = - 32 i = 0 - 32 i, ((1 + i)^{2})^{5} = 32 i = 0 + 32 i .

Thus the real part is zero in each case, and the imaginary parts are

- 32

and

32

. The results are complex conjugates, as also follows because the two inputs are complex conjugates and the fifth-power polynomial has real coefficients.

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