Real number raised to complex number?

7^(10 + 8i) = 7^10 * 7^(8i)

7^10 = 282,475,249

7^(8i) = [e^(ln 7)]^(8i)

= e^[8 (ln 7) i]

= cos(8 ln 7) + i sin(8 ln 7)

= approx. cos(15.57) + i sin(15.57)

= approx. 0.9633 + 0.2684 i

So 7^(10 + 8i) = approx. 282,475,249 * (0.9633 + 0.2684 i)

= approx. 272,112,920 + 75,807,815 i

(10 + 7i)^2 = 100 - 49 + 140i = 51 + 140i

(10 + 7i)^4 = (51 + 140i)^2 = - 16,999 + 14,280i

(10 + 7i)^8 = (- 16,999 + 14,280i)^2

= 85,047,601 - 485,491,440i

(10 + 7i)^(13 - 8i)

= (10 + 7i)^13 / (10 + 7i)^(8i)

(10 + 7i)^13 can be evaluated as in question 2.

10^2 + 7^2 = 100 + 49 = 149

arctan(7/10) = approx. 0.6107

so 10 + 7i = approx. (sqrt 149) e^(0.6107i)

and (10 + 7i)^(8i) = approx. [(sqrt 149) e^(0.6107i)]^(8i)

= approx. (sqrt 149)^(8i) * e^(- 0.6107 * 8)

(sqrt 149)^(8i) can be evaluated by a procedure similar to that of the question 1.

e^(- 0.6107 * 8) is a real number.

So (10 + 7i)^(13 - 8i) can be evaluated, but I think I'll leave it to you to find its value!

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