Final answer:
To find all the roots of (−√7+i)¹/² using De Movier's Theorem, we need to express the number in polar form and then find the values of the roots.
Step-by-step explanation:
To find all the roots of (−√7+i)¹/² using De Movier's Theorem, we need to express the number in polar form and then find the values of the roots. Let's start by converting −√7+i into polar form. The magnitude (r) can be found using the formula |z| = √(a² + b²), where a is the real part and b is the imaginary part. In this case, a = −√7 and b = 1. Therefore, |z| = √((-√7)² + 1²) = √(7 + 1) = √8 = 2√2.
The argument (θ) can be found using the formula tan(θ) = b/a. In this case, θ = tan⁻¹(1/(-√7)). Using a calculator, we find that θ ≈ 41.19°.
Now, we can express the number in polar form as z = 2√2(cos(41.19°) + i sin(41.19°)).
According to De Movier's Theorem, the nth root of a complex number can be found by taking the nth root of the magnitude and dividing the argument by n. In this case, we need to find the square root, so n = 2. The two square roots of z can be found as follows:
The first square root is given by ∛(2√2)(cos(41.19°/2) + i sin(41.19°/2)).
The second square root is given by ∛(2√2)(cos((41.19° + 360°)/2) + i sin((41.19° + 360°)/2)).