(x + 1/x)^2 = x^2 + 2 + 1/x^2 = x^2 + 1/x^2 + 2
We know that x^2 + 1/x^2 = 14, thus (x + 1/x)^2 = 16. Then x + 1/x is either 4 or -4. It can't be -4 because that would imply that x is negative. So x + 1/x = 4. Then raising that to the fifth power gives (x + 1/x)^5 = 1024.
We know that x^2 + 1/x^2 = 14, thus (x + 1/x)^2 = 16. Then x + 1/x is either 4 or -4. It can't be -4 because that would imply that x is negative. So x + 1/x = 4. Then raising that to the fifth power gives (x + 1/x)^5 = 1024.
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From the first equation, x = +/-2 +/- sqrt[ 3]. Since 0 < x, we use just the two positive values. So then, using either value of x, the second expression is equal to 1024.
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x² + 1/x² = 14
x² + 2 + 1/x² = 16
(x + 1/x)² = 16
x + 1/x = 4
(x + 1/x)⁵ = 4⁵ = 1024
x² + 2 + 1/x² = 16
(x + 1/x)² = 16
x + 1/x = 4
(x + 1/x)⁵ = 4⁵ = 1024