If $\displaystyle z = \frac{\sqrt{3}+1}{2}$ find the value of $\displaystyle z^{69}$

Printable View

- May 3rd 2009, 09:45 PMzorroFind the value of Z^69
If $\displaystyle z = \frac{\sqrt{3}+1}{2}$ find the value of $\displaystyle z^{69}$

- May 4th 2009, 12:03 PMvemrygh
Write the complex number on the form $\displaystyle z=re^{\theta i}$, and then compute $\displaystyle z^{69}=r^{69}e^{69\theta i}.$

- May 4th 2009, 01:30 PMSoroban
Hello, zorro!

Quote:

$\displaystyle \text{If }z = \frac{\sqrt{3}+1}{2}, \text{ find the value of }z^{69}$

Convert $\displaystyle z$ to polar form: .$\displaystyle z \:=\:\cos\tfrac{\pi}{6} + i\sin\tfrac{\pi}{6}$

Then use DeMoivre's Theorem: .$\displaystyle (\cos\theta + i\sin\theta)^n \;=\;\cos(n\theta) + i\sin(n\theta)$

- May 5th 2009, 12:49 AMCalculus26
Am I missing something here?

I would agree with y'all if it were

[3^(1/2)+i]/2

But as z is not complex?????? - May 5th 2009, 05:16 AMmr fantastic
- Jun 28th 2009, 02:18 AMzorroAnswer please
i have tried to solve the problem but dont know what is the right answer .Could u please let me know what is the right answer .........

- Jun 28th 2009, 03:07 AMPlato
- Jun 28th 2009, 03:29 AMaidan
$\displaystyle z = \frac{\sqrt{3}+1}{2}$ = 1.366025404 approx.

find the value of $\displaystyle z^{69}$

$\displaystyle {1.366025404}^{69} = 2221547172.+decimaldigits$

How many decimals places are needed.

or do you need the binominal expansion of this

$\displaystyle \frac{ \left ( \sqrt{3}+1 \right )^{69} } {2^{69}}$ - Jun 29th 2009, 11:47 PMzorroWant to know the right solution
If $\displaystyle z=\frac{\sqrt{3}+1}{2}$ , find the value of $\displaystyle z^{69}$

- Jun 30th 2009, 12:37 AMMoo
Okay, let me explain it to you :

In the form you've written, it is not possible to get a nice form for the exact value !

If it were $\displaystyle \frac{\sqrt{3}+i}{2}$, then the problem would be less meaningless.

Consider again your problem. - Jun 30th 2009, 12:53 AMmr fantastic
There is no simple

**exact**answer to the question you have posted. This has been said several times now.

So either the question has a mistake in it or an approximate answer is required. If the former, then the likely question and its answer has been given in this thread. If the latter, the answer has also been given.

I don't see how any further progress can be made here unless the OP is prepared to acknowledge and address the above issues. - Jul 4th 2009, 01:46 PMaidan
The

answer:__EXACT__

$\displaystyle \frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712} $ - Jul 4th 2009, 02:02 PMPlato
As you can see your exact solution is a bit off.

Go to Wolfram|Alpha. Type “((sqrt[3]+1)/2)^69” exactly then click the “=” at the right-hand side of the input box. - Jul 4th 2009, 08:53 PMCaptainBlack
- Jul 4th 2009, 11:54 PMzorroSorry for the typo mistake