Algebra Tutorials!
Sunday 14th of April  
Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Angles and Degree Measure
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Roots of Complex Numbers

Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial equation of degree n has n solutions in the complex number system. Each solution is an nth root of the equation. The nth root of a complex number is defined as follows.


Definition of nth Root of a Complex Number

The complex number u = a + bi is an nth root of the complex number z if z = un = (a + bi)n.



The nth roots of a complex number are useful for solving some polynomial equations. For instance, you can use DeMoivre’s Theorem to solve the polynomial equation x4 + 16 = 0 by writing -16 as 16(cos π + i sin π).


To find a formula for an nth root of a complex number, let u be an nth root of z, where u = s(cos β + i sin β) and z = r(cos θ + i sin θ).

By DeMoivre’s Theorem and the fact that un = z  you have

sn(cos nβ + i sin nβ) = r (cos θ + i sin θ).

Taking the absolute values of both sides of this equation, it follows that sn = r. Substituting back into the previous equation and dividing by r, you get

cos nβ + i sin nβ = cos θ + i sin θ

Thus, it follows that

cos nβ = cos θ and sin nβ = sin θ.

Because both sine and cosine have a period of 2π, these last two equations have solutions if and only if the angles differ by a multiple of 2π. Consequently, there must exist an integer k such that

By substituting this value for into the polar form of u, you get the following result.


When k exceeds n - 1 the roots begin to repeat. For instance, if k = n the angle is coterminal with θ/n which is also obtained when k = 0.



nth Roots of a Complex Number

For a positive integer n, the complex number z = r(cos θ + i sin θ) has exactly n distinct nth roots given by

where k = 0, 1, 2, ..., n - 1.

This formula for the nth roots of a complex number z has a nice geometrical interpretation, as shown in the figure below.

Note that because the nth roots of z all have the same magnitude they all lie on a circle of radius with center at the origin. Furthermore, because successive nth roots have arguments that differ by 2π/n, the n roots are equally spaced along the circle.


Example 1

Finding the nth Roots of a Complex Number

Find the three cube roots of z = -2 + 2i.


Because z lies in Quadrant II, the polar form for z is

By the formula for nth roots, the cube roots have the form

Finally, for k = 0, 1, and 2, you obtain the roots

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