Let's give this problem a try. So in this problem, we are told if \( z \) is equal to \( 1024 \times \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) \), calculate the 5th root of \( z \). And recall that the 5th root of \( z \) is the same thing as \( z^{1/5} \). Now how we can solve this problem is by recognizing that we're dealing with a complex root. So for complex roots, what I'm going to do is write out everything that we know about these. Now I know that if I want to find \( Z^{1/n} \), that's going to equal the \( r \) value, the number out in front, to the \( 1/n \) power, and that's going to be times cis where cis is a combination of the cosines and sines it's gonna be cis \( \theta_k \). Now \( \theta_k \) is the tedious part of the problem where you have to calculate each of these angles, but I know that \( \theta_k \) is going to be \( 1/n \), and that's going to be that times our angle \( \theta \) plus \( 2\pi \times k \). Now since I can see that we have radians in this problem, I'm using radians, so \( 2\pi \). Now for the \( k \) values, the \( k \) values are going to start at 0 and it's gonna go 1, 2, all the way up to \( n-1 \). So those are gonna be the \( k \) values, this is \( \theta_k \), this is each of the \( z \) values, and that's the ultimate setup. Now what I can do is figure out in this problem what the \( r \) values are going to be and what the \( k \) values are going to be because that's going to help me a lot. So if I go over here, I can see that the \( r \) value is 1024. If I want to find \( r^{1/n} \), in this case, \( 1/n \) is \( 1/5 \), so you can see that this is gonna be equal to \( 1024^{1/5} \). Now this is something you can put into your calculator because this is gonna be pretty tedious to try and figure this out, but the 5th root of 1024 turns out to just be 4. So this is going to be the \( r \) value we use throughout this problem. Now next, what I'm going to do is figure out what our \( k \) values are because \( k \) values go from 0 all the way up to \( n-1 \). Well, I can see that \( n-1 \) is going to be 5 minus 1 this number of the denominator, and 5 minus 1 is 4. So that means our \( k \) values are going to be 0, 1, 2, 3, and 4, and these are the \( k \) values we're going to use in this problem. So as we have the \( r \) and we have the \( k \), so we can actually set up what this problem is going to look like. I can see that we have 1, 2, 3, 4, 5 \( k \) values, which means we're going to have 5 solutions in this problem. So we're going to have \( z_0 \), we're gonna have \( z_1 \), \( z_2 \). We're gonna have the \( z \) for each of these \( k \) values, so we're going to have go over here we'll have a \( z_3 \), and then we'll have \( z_4 \). And these are going to be all the solutions we have in this problem. Now for \( z_0 \), this is going to be this pattern right here. So we're going to have \( r^{1/n} \) which we said was 4. So we have \( 4 \) cis, and again, this is just the cosines and sines, and that's going to be cis some angle, and then for \( z_1 \) we're going to have \( 4 \) cis some angle, and this pattern just repeats. So here we're gonna have \( 4 \) cis some angle, and you can keep doing this for each of these solutions we get. Because the pattern stays the same, the \( r \) value stays the same, but it's just the angle inside that's going to change. Because what we're trying to do now from here is figure out what this \( \theta_k \) is, and that's really the tricky part of solving t
Table of contents
- 0. Fundamental Concepts of Algebra3h 29m
- 1. Equations and Inequalities3h 27m
- 2. Graphs1h 43m
- 3. Functions & Graphs2h 17m
- 4. Polynomial Functions1h 54m
- 5. Rational Functions1h 23m
- 6. Exponential and Logarithmic Functions2h 28m
- 7. Measuring Angles39m
- 8. Trigonometric Functions on Right Triangles2h 5m
- 9. Unit Circle1h 19m
- 10. Graphing Trigonometric Functions1h 19m
- 11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
- 12. Trigonometric Identities 2h 34m
- 13. Non-Right Triangles1h 38m
- 14. Vectors2h 25m
- 15. Polar Equations2h 5m
- 16. Parametric Equations1h 6m
- 17. Graphing Complex Numbers1h 7m
- 18. Systems of Equations and Matrices1h 6m
- 19. Conic Sections2h 36m
- 20. Sequences, Series & Induction1h 15m
- 21. Combinatorics and Probability1h 45m
- 22. Limits & Continuity1h 49m
- 23. Intro to Derivatives & Area Under the Curve2h 9m
17. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
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