cube root of i, feat. Fematika

TL;DR

Understanding the multiple solutions to the cube root of i and exploring the correct approach to solve it.

Transcript

hello I'm Monica and in today's video I will be evaluating the cube root of I there multiple valued so let's figure out all of the solutions first using the horrible way you should never do and then second using the way you should always do so let's set this equal to some generic complex number A plus bi then cube both sides and I get I on this sid... Read More

Key Insights

• 😃 Using the incorrect method, the solutions a = (√3/2) and b = 1/2 are obtained.
• 🥺 The incorrect method can lead to complex solutions and is not reliable.
• 🥳 The correct method involves separating the equation into real and imaginary parts to solve for a and b.
• 💁 The alternative method using complex exponential form provides additional unique solutions.
• 🫚 The final solutions to the cube root of i are (√3/2 + 1/2i), (-√3/2 + 1/2i), and -i.
• 💁 The cube root of i can be represented in exponential form as e^(iπ/6), e^(5iπ/6), and e^(3iπ/2).
• 😑 The cube root solutions can also be expressed as cosine and sine functions.

Questions & Answers

Q: What are the two methods discussed in the video to solve the cube root of i?

The two methods are the incorrect method, which involves separating the equation into real and imaginary parts, and the correct method, which uses complex exponential form.

Q: How are the solutions derived by separating the equation into real and imaginary parts?

By setting the real part equal to 0, the equation a³ - 3aB² = 0 is obtained. Similarly, setting the imaginary part equal to 1 leads to the equation 3a²B - b³ = 1. Solving these equations simultaneously provides the solutions for a and b.

Q: What is the alternate method discussed using complex exponential form?

The alternative method involves expressing i as e^(iπ/2), and then applying the cube root. By adjusting the value of n, unique solutions are obtained.

Q: What are the final solutions to the cube root of i?

The solutions are: (√3/2 + 1/2i), (-√3/2 + 1/2i), and -i.

Summary & Key Takeaways

• The video discusses two methods to solve the cube root of i - the incorrect way and the correct way.

• By separating the equation into real and imaginary parts, the solution is derived.

• Another approach, using complex exponential form and applying the cube root, provides additional unique solutions.