# Algebra problem (x+(1/x) problem)

1. May 18, 2016

### terryds

1. The problem statement, all variables and given/known data

If $a+\frac{1}{a}=\sqrt{3}$, then $a^{2016} + (\frac{1}{a})^{2016}$ equals

A. 0
B. 1
C. √2
D. √3
E. 2

2. Relevant equations

3. The attempt at a solution

I find no real solution for a.
What I can do is just find a^2 + (1/a)^2 which equals 1, a^4 + (1/a)^4 = -1, a^8 + (1/a)^8 = -1, and so on...
Since 2016 is not a square number, I don't know how to determine the value

Is there any theory about problem like this?

2. May 18, 2016

### Samy_A

Start from $a²+1/{a²}=1$.
Multiply by $a²$ to get rid of the denominator, you get $a^4 +1 =a²$.
Now notice that 2016 is a multiple of 6. So multiply $a^4 +1 =a²$ again by $a²$ to get $a^6$ in the equation.

3. May 18, 2016

### ehild

Find the complex solution. Write it in trigonometric of exponential form, which makes it easy to rise to power 2016. (2016=25*32 *7)

4. May 18, 2016

### terryds

a^6 + a^2 = a^4
a^6 = a^4 - a^2 = (a^2 - 1) - a^2 = -1
a^12 = 1
a^24 = 1
...
a^2016 = 1

So, 1/a^2016 = 1
a^2016 + (1/a)^2016 = 1 + 1 = 2
Thank you

5. May 19, 2016

### terryds

I'm interested to try this approach..

The solution for a is $\frac{\sqrt{3}\pm i}{2}$
Should I use the plus or the minus as the root?
If it's plus, $e^{i\frac{1}{6}\pi}$
If it's minus, $e^{i\frac{11}{6}\pi}$

And, then what should I do??

6. May 19, 2016

### ehild

What are their 12th power?

7. May 19, 2016

### terryds

The 'plus' root^12 = $e^{i2\pi}= 1$
The 'minus' root^12 = $e^{i22\pi} = 1$

Okay, since 2016 can be divided by 12, a^2016 = 1
And 1/a^2016 = 1
So, a^2016 + 1/a^2016 = 1+1 = 2

Thanks a lot!

8. May 19, 2016

### ehild

The 'minus' root^12 = $e^{-i2\pi} = 1$