Final answer:
To find the indicated power of (√3+i)⁻⁴ using Moivre's theorem, convert (√3+i) into polar form. Apply Moivre's theorem to raise it to the -4th power. Evaluate the expression to get the final answer.
Step-by-step explanation:
To find the indicated power of (√3+i)⁻⁴ using Moivre's theorem, we first need to convert (√3+i) into polar form. The magnitude of (√3+i) is √(√3² + 1²) = 2, and the angle is arctan(1/√3) = π/6. Therefore, (√3+i) can be written as 2(cos(π/6) + i*sin(π/6)).
Using Moivre's theorem, we can then raise 2(cos(π/6) + i*sin(π/6)) to the -4th power. Applying the theorem, we get 2⁻⁴ (cos(-4*π/6) + i*sin(-4*π/6)). Simplifying further, we have 2⁻⁴ (cos(-2π/3) + i*sin(-2π/3)).
Finally, we evaluate the expression. 2⁻⁴ = 1/16, and the cosine and sine of -2π/3 are -1/2 and -√3/2 respectively. Therefore, (√3+i)⁻⁴ = (1/16)(-1/2 - (√3/2)i).