An algebra problem by Md Zuhair

Algebra Level 3

If x + 1 x = 1 x + \dfrac{1}{x} = 1 , then find the value of x 12 x^{12} .

0 3 1 4

Md Zuhair by Md Zuhair , 20, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Chew-Seong Cheong
Jul 19, 2016

x + 1 x = 1 x 2 − x + 1 = 0 x 2 = x − 1 ⟹ x 3 = x 2 − x = x − 1 − x ⟹ x 3 = − 1 x 12 = ( x 3 ) 4 = ( − 1 ) 4 = 1 \begin{aligned} x + \frac 1x & = 1 \\ x^2 - x + 1 & = 0 \\ \color{#3D99F6}{x^2} & \color{#3D99F6}{= x - 1} \\ \implies x^3 & = \color{#3D99F6}{x^2} - x \\ & = \color{#3D99F6}{x-1} - x \\ \implies \color{#D61F06}{x^3} & \color{#D61F06}{= - 1} \\ x^{12} & = (\color{#D61F06}{x^3})^4 \\ & = (\color{#D61F06}{-1})^4 \\ & = \boxed{1} \end{aligned}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Good job..

Md Zuhair - 4 years, 11 months ago
Steven Chase
Jul 20, 2016

Here's a different take:

2 Helpful 0 Interesting 0 Brilliant 1 Confused
Md Zuhair
Jul 19, 2016

Easy .. Find the value of x which will be equal to w .. where w is the cube root of unity ,,, Hence (w)12= (w^3)4 = 1^4 = 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...