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Algebra Level 3

If x + 1 x = 2 cos ⁡ ( π 4 ) \large x + \frac{1}{x} = 2\cos\left(\frac{\pi}{4}\right)

then, x 12 = ? x^{12} = \,?


The answer is -1.

Naren Bhandari by Naren Bhandari , 21, India

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2 solutions

x + 1 x = 2 cos ⁡ π 4 = 2 ( x + 1 x ) 2 = ( 2 ) 2 x 2 + 2 + 1 x 2 = 2 x 2 + 1 x 2 = 0 ⟹ x 2 = − 1 x 2 x 4 = − 1 ( x 4 ) 3 = ( − 1 ) 3 ⟹ x 12 = − 1 \begin{aligned} x+\frac 1x & = 2 \cos \frac \pi 4 = \sqrt 2 \\ \left(x+\frac 1x\right)^2 & = (\sqrt 2)^2 \\ x^2 + 2 + \frac 1{x^2} & = 2 \\ x^2 + \frac 1{x^2} & = 0 \\ \implies x^2 & = - \frac 1{x^2} \\ x^4 & = -1 \\ (x^4)^3 & = (-1)^3 \\ \implies x^{12} & = \boxed{-1} \end{aligned}

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Wow! Nice solution, sir.

Majed Kalaoun - 4 years ago
Harry Jones
Jun 4, 2017

cos ⁡ π 4 = ( x + 1 x ) 2 \cos\dfrac{\pi}{4}=\dfrac{\left(x+\dfrac{1}{x}\right)}{2}

x=e^\left(\dfrac{i\pi}{4}\right)

x 12 = e i 3 π = cos ⁡ 3 π + i sin ⁡ 3 π = − 1 \begin{aligned}x^{12}=e^{i3\pi}=\cos3\pi+i\sin3\pi=-1\end{aligned}

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