In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 1 + i

Verified step by step guidance

1. Express the complex number 1 + i in polar form. Calculate the modulus: $r=\sqrt{1^2+1^2}=\sqrt{2}$.
2. Determine the argument (angle) $\theta$ using ${\tan }^{-1}(\frac{1}{1})=\frac{\pi }{4}$ radians or 45 degrees.
3. Use De Moivre's Theorem to find the sixth roots: $z_k=\sqrt[6]{r}(\cos \left(\frac{\theta +2k\pi }{6}\right)+i\sin \left(\frac{\theta +2k\pi }{6}\right))$ for $k=0,1,2,3,4,5$.
4. Calculate $\sqrt[6]{\sqrt{2}}$ for the modulus of each root.
5. Substitute $k=0,1,2,3,4,5$ into the formula to find each root in rectangular form, rounding to the nearest tenth if necessary.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In this context, 1 + i is a complex number where the real part is 1 and the imaginary part is 1. Understanding how to manipulate and represent complex numbers is essential for finding their roots.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ) with respect to the positive real axis, represented as r(cos θ + i sin θ) or re^(iθ). This form is particularly useful for finding roots of complex numbers, as it simplifies the calculations involved in extracting roots, especially when dealing with multiple roots.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form and any integer n, the nth roots can be found using the formula: r^(1/n)(cos(θ/n + 2kπ/n) + i sin(θ/n + 2kπ/n)), where k is an integer from 0 to n-1. This theorem is crucial for calculating the complex roots of a number, as it provides a systematic way to derive all possible roots.