I lost my count!

Algebra Level 2

Given that 1 x = 2 − x \frac{1}{x}=2-x . x ≠ 0 x\neq 0

Evaluate the expression x 2 2014 + 1 x 2 2014 x^{2^{2014}}+\frac{1}{x^{2^{2014}}} .


The answer is 2.

Ahmad Naufal Hakim by Ahmad Naufal Hakim , 23, Indonesia

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3 solutions

Rewriting the equation, x 2 − 2 x + 1 = 0 x^{2}-2x+1=0

( x − 1 ) 2 = 0 ⟹ x = 1 (x-1)^{2}=0\implies x=1

Hence, x 2 2014 + 1 x 2 2014 = 2 \boxed{x^{2^{2014}}+\frac{1}{x^{2^{2014}}}=2}

10 Helpful 0 Interesting 0 Brilliant 0 Confused

This the greatest solution my eyes have ever had the privilege to witness.

Refath Bari - 4 years, 5 months ago
Ashish Menon
Jun 2, 2016

1 x = 2 − x x + 1 x = 2 x 2 + 1 x = 2 x 2 + 1 = 2 x x 2 − 2 x + 1 = 0 ( x − 1 ) 2 = 0 x − 1 = 0 x = 1 ⟹ x 2014 + 1 x 2014 = 1 2014 + 1 1 2014 x = 2 \dfrac{1}{x} = 2 - x\\ x + \dfrac{1}{x} = 2\\ \dfrac{x^2 + 1}{x} = 2\\ x^2 + 1 = 2x\\ x^2 - 2x + 1 = 0\\ {(x - 1)}^2 = 0\\ x - 1 = 0\\ x = 1\\ \implies x^{2014} + \dfrac{1}{x^{2014}} = 1^{2014} + \dfrac{1}{1^{2014}}\\ x = \color{#69047E}{\boxed{2}} .

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Look again at the problem!

Eman Shaban - 4 years, 5 months ago

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Oops it's my own fault

Eman Shaban - 4 years, 5 months ago
Finn Hulse
Apr 15, 2014

Isn't it a bit obvious that x = 1 x=1 ? I don't know about you guys, I just saw that immediately. So when you evaluate the second bit, you get 1 + 1 = 2 1+1=\boxed{2} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It is obvious, but I wrote the equation like that on purpose, although it was useless actually.

Ahmad Naufal Hakim - 7 years, 1 month ago

Same with me Finn

ashutosh mahapatra - 7 years, 1 month ago

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