Algebraic Manipulation

Algebra Level pending

If x + 1 x = 5 x+\frac{1}{x}=5 , then what is x 4 + 1 x 4 x^4+\frac{1}{x^4} ?

625 623 561 527

Keshav Ramesh by Keshav Ramesh , 18, USA

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1 solution

Zee Ell
Feb 12, 2017

x + 1 x = 5 x + \frac {1}{x} = 5

( x + 1 x ) 2 = x 2 + 1 x 2 + 2 × x × 1 x = x 2 + 1 x 2 + 2 (x + \frac {1}{x} )^2 = x^2 + \frac {1}{x^2} + 2 × x × \frac {1}{x} = x^2 + \frac {1}{x^2} + 2

x 2 + 1 x 2 = ( x + 1 x ) 2 − 2 = 5 2 − 2 = 23 x^2 + \frac {1}{x^2} = ( x + \frac {1}{x} )^2 - 2 = 5^2 - 2 = 23

x 4 + 1 x 4 = ( x 2 + 1 x 2 ) 2 − 2 = 2 3 2 − 2 = 527 x^4 + \frac {1}{x^4} = ( x^2 + \frac {1}{x^2} )^2 - 2 = 23^2 - 2 = \boxed {527}

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