Find the powers of each complex number in polar form. Find z^4 when z=2 cis(70^∘).

Name: Problem 347, Chapter 8: Precalculus
Uploaded: 2020-02-27T04:49:28-08:00
Duration: 1 min 21 s
Channel: Jim Van Der Valk Bouman
Description: Find the powers of each complex number in polar form. Find z^4 when z=2 cis(70^∘).

step 1 In this question we are given a complex number Z is 2 s 70 degrees. And we have to calculate wha

Find the powers of each complex number in polar form. Find...

Key Concepts

Exponentiation of Complex Numbers

Exponentiation of complex numbers involves raising a complex number to a power, which, when the number is in polar form, translates to raising the modulus to that power and multiplying the argument by the exponent. This process is streamlined by using De Moivre's Theorem and is central to solving many problems in complex analysis and trigonometry.

De Moivre's Theorem

De Moivre's Theorem is a key result in complex number theory that allows for the efficient computation of powers and roots of complex numbers in polar form. It states that (r cis(?))^n = r^n cis(n?), which means that when a complex number is raised to the nth power, the magnitude is raised to the nth power and the angle is multiplied by n.

Complex Numbers

Complex numbers consist of a real part and an imaginary part, and they can be represented graphically in the complex plane. This representation allows complex numbers to be expressed in different forms, such as rectangular (a + bi) and polar form, which facilitates various types of calculations and geometric interpretations.

Polar Form

The polar form of a complex number expresses it in terms of its magnitude (or modulus) and an angle (or argument) from the positive real axis. It is typically written as r cis(?), where r is the magnitude and cis(?) stands for cos(?) + i sin(?). This form is especially useful for multiplying and raising complex numbers to a power.

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