In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argumen

Step 1: Express u005C( i u005C) in polar form.u000AThe complex number u005C( i u005C) can be written as u005C( 0 + 1i u005C)

In Problems 55-62, find all the complex roots. Leave your...

Key Concepts

Polar Form Representation

This concept involves expressing a complex number in the form r(cos ? + i sin ?), where r is the modulus (or absolute value) representing the distance from the origin, and ? is the argument (or angle) which indicates the direction of the complex number in the complex plane. It simplifies operations like multiplication, division, and finding powers and roots of complex numbers.

Modulus and Argument

The modulus r of a complex number gives its magnitude, while the argument ? provides the angle (typically measured in degrees or radians) that the line representing the complex number makes with the positive real axis. These components are essential for converting a complex number from its standard form to its polar form.

De Moivre's Theorem

De Moivre's Theorem is a fundamental principle in complex number theory which states that for any complex number in polar form and any integer n, raising the complex number to the power n multiplies the argument by n and raises the modulus to the n. This theorem is particularly useful when calculating roots of complex numbers as it neatly relates exponentiation to trigonometric functions.

nth Root of a Complex Number

Finding the nth roots of a complex number involves applying a general formula that takes the nth root of the modulus and divides the argument by n, while also accounting for the multiple possible values (usually expressed with an added term of 360°k/n, where k is an integer) that arise due to the periodic nature of the trigonometric functions. This results in n distinct roots evenly distributed in the complex plane.

In Problems 55-62, find all the complex roots. Leave your answers in polar form with the argument in degrees. The complex fifth roots of $i$

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