Question

Solve each equation. $$z^{6}-64 i=0$$

   Solve each equation.
$$z^{6}-64 i=0$$

Higher Level Mathematics

Higher Level Mathematics

Ibrahim Wazir, Tim… 2012 Edition

Chapter 10, Problem 36

Instant Answer

Solved by Expert Varsha Aggarwal

01/13/2022

Step 1

We know that $64i = 64e^{i\frac{\pi}{2}}$ because $64i$ is a complex number with magnitude $64$ and angle $\frac{\pi}{2}$. So, the equation becomes $z^6 = 64e^{i\frac{\pi}{2}}$. Show more…

Show all steps

lock

Thanks for your feedback!

Solve each equation. $$z^{6}-64 i=0$$

Play audio

Feedback

Powered by NumerAI

Varsha Aggarwal and 78 other subject Algebra educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Root Extraction of Complex Numbers

Extracting the nth roots of a complex number involves expressing the number in polar form and then finding all distinct solutions given by z = r^(1/n) · e^(i(? + 2?k)/n), where k ranges from 0 to n-1. This method systematically provides all the roots and highlights the symmetrical nature of these solutions in the complex plane.

De Moivre's Theorem

De Moivre's Theorem asserts that (r · e^(i?))^n = r^n · e^(i n?) for any integer n, making it a fundamental tool for raising complex numbers to powers and extracting roots. This theorem is essential in solving equations like z^n = w, where taking the nth root involves dividing the argument by n and considering all possible angles that differ by multiples of 2?/n.

Complex Numbers

Complex numbers are numbers expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. They extend real numbers and enable the solution of equations that have no real solutions, thus playing a crucial role in many areas of mathematics, including algebra, calculus, and engineering.

Polar Representation

The polar representation of a complex number expresses it in the form r(cos? + i sin?) or r · e^(i?), where r is the modulus (absolute value) and ? is the argument (angle). This form is particularly useful when multiplying, dividing, or extracting roots of complex numbers, as it simplifies the operations by turning them into manipulations of the modulus and the angle.

Euler's Formula

Euler's formula, stated as e^(i?) = cos? + i sin?, serves as a bridge between exponential functions and trigonometric functions. It not only underpins the polar form of complex numbers but also simplifies many operations, such as exponentiation and finding roots in the complex plane.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later