GRE Math Subject Practice Test #43

In summary, the conversation discusses a problem involving complex analysis and seeks help in solving it. The problem involves finding the value of an expression with a complex number, and the conversation suggests using trigonometric functions and exponential form to solve it. The conversation also mentions that there may be shortcuts to solving the problem.
  • #1
Zabopper
2
0
First of all, I hope I'm posting this in the appropriate forum and let me know if I'm not.

I did pretty okay on this practice test, but I never took complex analysis, though I've tried to teach myself the rudiments. Maybe that's not even the problem, but this seems kind of basic, something I shouldn't be missing. So I'm looking for help with how to solve this problem, and also suggestions for preparing for the exam so that I don't miss this category of problem come test day. Here it is:

If z = e^(2*pi *i / 5), then 1 + z + z^2 + z^3 +5*z^4 + 4*z^6 + 4*z^7 +4*z^8 +5*z^9 =

a)0

b) 4*e^(3*pi*i / 5)

c) 5*e^(4*pi*i / 5)

d) -4*e^(2*pi*i / 5)

e) -5*e^(3*pi*i / 5)

Thank you everyone!
 
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  • #2
[tex] e^{xi}=cosx+isinx[/tex] NOw u have:

[tex] z= e^{{2\pi}\frac{i}{5}}=cos\frac{2\pi}{5} +isin\frac{2\pi}{5}[/tex]

[tex] z^2=e^{2*{2\pi}\frac{i}{5}}=cos\frac{4\pi}{5}+isin\frac{4\pi}{5}[/tex]

Now u can play a lill bit with trig functions of a double angle, so some things will cancel out. Also

[tex] 1=e^{{2\pi}i}=cos2\pi+isin{2\pi}[/tex]

i think that doing this for the whole powers of z and looking for a pattern of how things will cancle out, you should be able to get to the result.
 
  • #3
There might be shortcuts though, but none of which i can think at the moment!
 
  • #4
I think that's the hard way to do it. It is much simpler to do the multiplications in exponential form than "cos + i sin".

If [tex]z= e^{2\pi i/5}[/itex] then
[tex]z^2= e^{4\pi i/5}[/tex]
[tex]z^3= e^{6 pi i/5}= e^{pi i}e^{\pi i/5}= -e^{\pi i/5}[/tex]
[tex]z^4= e^{8\pi i/5}= -e^{3\pi/5}[/tex]
[tex]z^5= e^{10\pi i/5}= e^{2\pi i}= 1[/tex]
[tex]z^6= e^{12\pi i/5}= e^{2\pi i}e^{2\pi i/5}= e^{2\pi i/5}[/tex]
[tex]z^7= e^{14\pi i/5}= e^{2\pi i}e^{4\pi i/5}[/tex]
[tex]z^8= e^{16\pi i/5}= e^{3\pi i}e^{\pi i/5}= -e^{\pi i/5}[/tex]
and
[tex]z^9= e^{18\pi i/5}= e^{3\pi i}e^{3\pi i/5}= -e^{3\pi i/5}[/tex]

You should be able to put those into your formula and come up with an answer.
 

1. What topics are covered on the GRE Math Subject Practice Test #43?

The GRE Math Subject Practice Test #43 covers topics such as algebra, geometry, calculus, and data analysis. It also includes questions related to number theory, discrete mathematics, and probability.

2. Is the GRE Math Subject Practice Test #43 similar to the actual GRE Math Subject Test?

Yes, the GRE Math Subject Practice Test #43 is designed to simulate the format and content of the actual GRE Math Subject Test. It is a useful tool for familiarizing yourself with the types of questions and level of difficulty you can expect on the real test.

3. How can I use the GRE Math Subject Practice Test #43 to prepare for the actual test?

The GRE Math Subject Practice Test #43 can be used as a study tool to identify areas of strength and weakness in your math skills. By reviewing the questions you answered incorrectly, you can focus your study efforts on those topics.

4. How long does it take to complete the GRE Math Subject Practice Test #43?

The GRE Math Subject Practice Test #43 is a full-length test with 66 questions and a time limit of 170 minutes. It is recommended to take the test in one sitting, simulating the conditions of the actual test, but you can also break it up into smaller sections if needed.

5. Is there an answer key available for the GRE Math Subject Practice Test #43?

Yes, an answer key is included with the GRE Math Subject Practice Test #43. It provides explanations for each answer choice and the correct answer for every question on the test.

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