Final answer:
To find the indicated power using DeMoivre's Theorem for the expression (1-i)⁸, we need to convert it to polar form, apply DeMoivre's Theorem, and then convert back to rectangular form. The result is 256.
Step-by-step explanation:
To find the indicated power using DeMoivre's Theorem for the expression (1-i)⁸, we first need to write it in polar form. The polar form of (1-i) can be found by representing it as z = r(cosθ + isinθ), where r is the magnitude of the complex number and θ is the argument or angle. The magnitude of (1-i) can be found using the Pythagorean theorem: magnitude (r) = √(1² + (-1)²) = √2. The angle θ can be found using arctan: θ = arctan(-1/1) = -π/4.
Now that we have the polar form, we can apply DeMoivre's Theorem which states that (r(cosθ + isinθ))ⁿ = rⁿ(cos(nθ) + isin(nθ)). In this case, (1-i)⁸ = (2(cos(-π/4) + isin(-π/4)))⁸. We can now raise the magnitude and angle to the power of 8: (2⁸)(cos(-8π/4) + isin(-8π/4)), which simplifies to 256(cos(-2π) + isin(-2π)).
Using Euler's formula e^(iθ) = cosθ + isinθ, we can rewrite the answer in rectangular form: 256(e^(i(-2π))) = 256(cos(-2π) + isin(-2π)) = 256(1 + 0i) = 256.