Question

PLEASE HELP ME
the complex number z has modulus 5 1/16 and argument 4pi/9
(a) find, in polar form the four complex fourth roots of z( that is, find the four values of w^4=z)

Answer 1

Chan rows at a rate of 8 mph in still water. It takes him three hours to row upstream from his house to the park. He rows back home, and it takes him two hours. What is the speed of the current?

Which of the following is a clue that would help solve this word problem?
The rate of Chan’s rowing in still water is 8 mph.
Chan rows from the house to the park.
Chan rows back from the park to the house.
The entire trip takes five hours.

Answer 2

why is this on my question ? haha

Answer 3

my bad all that for what

Answer 4

haha its fine :L i don't know the answer to yours tho sorry

Answer 5

Chan rows at a rate of 8 mph in still water. It takes him three hours to row upstream from his house to the park. He rows back home, and it takes him two hours. What is the speed of the current?

Which of the following is a clue that would help solve this word problem?
The rate of Chan’s rowing in still water is 8 mph.
Chan rows from the house to the park.
Chan rows back from the park to the house.
The entire trip takes five hours.

Answer 6

Hey Sam,
\(\Large\bf \color{#A57F02}{\text{Welcome to OpenStudy! :)}}\)

Answer 7

Am I reading the modulus correctly? It's a mixed number?
\[\Large\sf |z|=5\frac{1}{16}, \qquad \arg(z)=\frac{4\pi}{9}\]

Answer 8

So one way to write z is in exponential form:\[\Large\bf\sf z=r e^{i \theta}\qquad\to\qquad z=|z| e^{i \arg(z)}\]

Answer 9

Let's write z as a multi-valued function so we can properly find the roots,\[\Large\bf\sf z=5\frac{1}{16}e^{i\frac{4\pi}{9}+2k \pi i},\qquad k=0,\pm1,\pm2,...\]This is the angle 4pi/9 but we allow any multiple of 2pi to be added or subtracted to show it's the same value.

Answer 10

Finding the 4 roots of z,\[\Large\bf\sf z^{1/4}=\left(5\frac{1}{16}e^{i\frac{4\pi}{9}+2k \pi i}\right)^{1/4},\qquad k=0,1,2,3\]So the thing that happens here is,
We have to restrict our k values to only be the first 4 positive values.
This gives us the 4 unique roots of z, after that they'll just repeat.

Answer 11

Using rules of exponents,\[\Large\sf =\quad \left(5\frac{1}{16}\right)^{1/4}\cdot e^{i\frac{4\pi}{36}+\frac{2k \pi i}{4}},\qquad k=0,1,2,3\]Which simplifies to,\[\Large\sf =\quad \left(5\frac{1}{16}\right)^{1/4}\cdot e^{i\frac{\pi}{9}+\frac{k \pi i}{2}},\qquad k=0,1,2,3\]

Answer 12

Letting k=0 gives us the first root,\[\Large\sf =\left(5 \frac{1}{16}\right)^{1/4}\cdot e^{i\frac{\pi}{9}+0}\]

Answer 13

k=1, 2 and 3 give us the other 3 unique roots.
Hopefully that helps a lil bit :)
Lemme know if that was too confusing.
I have a feeling I maybe misread your modulus lol.