Solve the equation: [tex]$Z^4=-8-j 8 \sqrt{3}$[/tex] leaving your answer in the form a+bj
Answer (1)
Convert the complex number to polar form: Z = r e j θ .
Find the magnitude r = 16 and argument θ = − 3 2 π .
Apply De Moivre's theorem to find the fourth roots: Z k = r 1/4 e j ( 4 θ + 2 πk ) , where k = 0 , 1 , 2 , 3 .
Calculate the four roots: 3 − j , 1 + j 3 , − 3 + j , − 1 − j 3 .
The solutions are: 3 − j , 1 + j 3 , − 3 + j , − 1 − j 3
Explanation
Problem Analysis We are given the equation Z 4 = − 8 − j 8 3 , where Z is a complex number, and we need to find the solutions for Z in the form a + bj , where a and b are real numbers. This problem involves finding the fourth roots of a complex number.
Convert to Polar Form First, convert the complex number − 8 − j 8 3 to polar form r e j θ , where r = ( − 8 ) 2 + ( − 8 3 ) 2 and θ = arctan ( − 8 − 8 3 ) .
Calculate Magnitude and Argument Calculate the magnitude r and the argument θ :
r = ( − 8 ) 2 + ( − 8 3 ) 2 = 64 + 192 = 256 = 16
θ = arctan ( − 8 − 8 3 ) = arctan ( 3 ) . Since the complex number is in the third quadrant, we need to add π to the arctangent result. Thus, θ = − 3 2 π .
Polar Form Express the complex number in polar form: − 8 − j 8 3 = 16 e − j 3 2 π .
De Moivre's Theorem Find the fourth roots of the complex number using De Moivre's theorem. The fourth roots are given by Z k = r 1/4 e j ( 4 θ + 2 πk ) , where k = 0 , 1 , 2 , 3 .
Calculate the Roots Calculate the four roots Z 0 , Z 1 , Z 2 , Z 3 by substituting k = 0 , 1 , 2 , 3 into the formula:
For k = 0 : Z 0 = 1 6 1/4 e j ( 4 − 3 2 π + 2 π ( 0 ) ) = 2 e − j 6 π = 2 ( cos ( − 6 π ) + j sin ( − 6 π )) = 2 ( 2 3 − j 2 1 ) = 3 − j
For k = 1 : Z 1 = 1 6 1/4 e j ( 4 − 3 2 π + 2 π ( 1 ) ) = 2 e j 3 π = 2 ( cos ( 3 π ) + j sin ( 3 π )) = 2 ( 2 1 + j 2 3 ) = 1 + j 3
For k = 2 : Z 2 = 1 6 1/4 e j ( 4 − 3 2 π + 2 π ( 2 ) ) = 2 e j 6 5 π = 2 ( cos ( 6 5 π ) + j sin ( 6 5 π )) = 2 ( − 2 3 + j 2 1 ) = − 3 + j
For k = 3 : Z 3 = 1 6 1/4 e j ( 4 − 3 2 π + 2 π ( 3 ) ) = 2 e j 6 11 π = 2 ( cos ( 6 11 π ) + j sin ( 6 11 π )) = 2 ( 2 3 − j 2 1 ) = 1 − j 3
Final Answer The four roots are:
Z 0 = 3 − j Z 1 = 1 + j 3 Z 2 = − 3 + j Z 3 = − 1 − j 3
Examples
Complex numbers are used in electrical engineering to represent alternating currents and voltages. The impedance of a circuit, which is the opposition to the flow of current, is also a complex number. Solving equations involving complex numbers, like finding the roots of a complex number, is crucial in analyzing and designing electrical circuits. For example, determining the stability of an electrical system often involves finding the roots of a characteristic equation, which can be a complex number.
