Find the root of \((1+i)^{1 / 5}\) with the smallest argument. a) \(0.168+1.06 i\) c) \(1.06-0.168 i\) b) \(1.06+0.168 i\) d) \(0.168-1.06 i\)

A cornerstone in the study of complex numbers, particularly within the context of engineering, is De Moivre's Theorem. This elegant theorem bridges the gap between complex numbers and trigonometry, offering a powerful tool for computing powers and roots of complex numbers.

De Moivre's Theorem states that for a complex number expressed in polar form as z = r(cos(θ) + isin(θ)), where r is the magnitude and θ is the argument, the nth power of z is given by zⁿ = rⁿ(cos(nθ) + isin(nθ)). When we apply this to find roots, we consider the reciprocal exponent 1/n.

For instance, using De Moivre's Theorem, we can solve expressions like \( (1+i)^{1/5} \) by first converting to polar form and then applying the theorem to find the resultant magnitude and argument. By examining the smallest argument, engineers can establish the principal root among the multiple solutions that complex roots inherently offer.

Understanding the polar form of complex numbers is essential when dealing with complex arithmetic in engineering applications. The polar form offers a more intuitive grasp of multiplication, division, and finding powers and roots of complex numbers.

Any complex number can be represented as a point in a plane with its x-coordinate corresponding to the real part and y-coordinate to the imaginary part. This Cartesian form, z = a + bi, can be transformed into its polar equivalent characterized by a magnitude r and an angle θ, yielding the polar form z = r(cos(θ) + isin(θ)).

The magnitude r is calculated as the square root of the sum of the squares of the real and imaginary parts, while the argument θ is the arctangent of the imaginary part over the real part. This transformation to polar form is the first step in applying De Moivre's Theorem, which underpins the process of complex number manipulation.

The concept of roots extends naturally from real numbers to complex numbers, allowing for solutions to equations that would otherwise have no real solution. When it comes to complex numbers, every non-zero number has n distinct nth roots that are evenly spaced around the origin of the complex plane.

These roots can be found by expressing the complex number in polar form and then applying De Moivre's Theorem using a fractional exponent. After finding the magnitude of the root, we can calculate the angles by dividing the original argument by n and adding multiples of 2π to find the different roots.

This calculation was demonstrated in identifying the root with the smallest argument for \( (1+i)^{1/5} \), by systematically exploring the argument of potential solutions. Such methods are critical in fields like electrical engineering, where the concept of phase is central, and finding the principle root has practical implications.