# Thread: Find the value of Z^69

1. ## Find the value of Z^69

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

2. Write the complex number on the form $z=re^{\theta i}$, and then compute $z^{69}=r^{69}e^{69\theta i}.$

3. Hello, zorro!

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

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

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

4. 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??????

5. Originally Posted by Calculus26
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??????
Probably a typo. Clarification might be given in a few weeks.

6. ## Answer 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 .........

7. Originally Posted by zorro
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 .........
Is the number z complex or real?

Is it $\frac{\sqrt{3}+{\color{red}i}}{2}$ or just $\frac{\sqrt{3}+1}{2}$?

8. Originally Posted by zorro
If $z = \frac{\sqrt{3}+1}{2}$ find the value of $z^{69}$
$z = \frac{\sqrt{3}+1}{2}$ = 1.366025404 approx.

find the value of $z^{69}$
${1.366025404}^{69} = 2221547172.+decimaldigits$
How many decimals places are needed.

or do you need the binominal expansion of this
$\frac{ \left ( \sqrt{3}+1 \right )^{69} } {2^{69}}$

9. ## Want to know the right solution

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

10. 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 $\frac{\sqrt{3}+i}{2}$, then the problem would be less meaningless.

Consider again your problem.

11. Originally Posted by zorro
If $z=\frac{\sqrt{3}+1}{2}$ , find the value of $z^{69}$
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.

12. The EXACT answer:

$\frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712}$

13. Originally Posted by aidan
The EXACT answer:
$\frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712}$
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.

14. Originally Posted by aidan
The EXACT answer:

$\frac {479994729504490251817683255296 \times \sqrt{3+1}}{ 590295810358705651712}$
Originally Posted by Plato
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.
What is more interesting about aidan's answer (other than the obvious typo: $\sqrt{3+1}$ ) is why is it wrong (at least I think this is of interest).

Cb

15. ## Sorry for the typo mistake

Originally Posted by Plato
Is the number z complex or real?

Is it $\frac{\sqrt{3}+{\color{red}i}}{2}$ or just $\frac{\sqrt{3}+1}{2}$?

It is
$\frac{\sqrt{3}+i}{2}$
Now please can any one give me the right solution to this question?

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