If #x + 1/x = sqrt(3)# then find the value of #x^17 + 1/x^17# ?

2 Answers
George C.
Feb 21, 2018

#x^17+1/x^(17) = -sqrt(3)#

Explanation:

Given:

#x+1/x = sqrt(3)#

Note that:

#x + 1/x = 2 cos(pi/6)#
#color(white)(x + 1/x) = (cos(pi/6)+i sin(pi/6)) + (cos(-pi/6)+i sin(-pi/6))#

So by de Moivre's formula:

#x^17 + 1/x^17 = 2 cos((17pi)/6) = 2 cos((5pi)/6) = -sqrt(3)#

George C.
Feb 21, 2018

#x^17+1/x^17 = -sqrt(3)#

Explanation:

Given:

#x+1/x = sqrt(3)#

Square both sides to get:

#x^2+2+1/x^2 = 3#

Subtract #3# from both sides to get:

#x^2-1+1/x^2 = 0#

Mutiply both sides by #x^4+x^2# to get:

#x^6+1 = 0#

That is: #x# is a sixth root of #-1#

So

#x^17+1/x^17 = x^18/x + x/x^18#

#color(white)(x^17+1/x^17) = (-1)^3/x + x/(-1)^3#

#color(white)(x^17+1/x^17) = -(1/x+x)#

#color(white)(x^17+1/x^17) = -sqrt(3)#

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