Final answer:
To find (1+i)⁸, express it in polar form and use De Moivre's theorem to raise (√2)⁸ to the power of e^(i * 8 * π/4). Simplify the expression to get the result.
Step-by-step explanation:
The given equation (x-4)²+y²=16 represents a circle with its center at (4,0) and a radius of 4. To find (1+i)⁸, we can express it in polar form and then use De Moivre's theorem. In polar form, 1+i has a magnitude (√2) and an angle of 45 degrees (π/4 radians) with the positive real axis. Applying De Moivre's theorem, we can raise (√2)⁸ to the power of e^(i * 8 * π/4) to get the desired result. Simplifying the expression, we have (√2)⁸ * cos(8 * π/4) + i * (√2)⁸ * sin(8 * π/4).