If X^2+1/X^2=7, then find the value of X^3-1/X^3

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Solution

X²+1/x² =7
(x+1/x)²-2=7
x +1/x =3

x³+1/x³ = (x+1/x)(x²+1/x² -1)
=3×(7-1)=18

or

(x + 1/x) = x + 1/x + 2
(x + 1/x) = 9
x + 1/x = 3
now
(x + 1/x) = x + 1/x + 3(x + 1/x)
27 = x + 1/x + 9
or
x + 1/x = 27 - 9 = 18

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