Answer: 110
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Work Shown:
Let's square both sides of the given equation and do a bit of algebra like so.
x+1/x = 5
(x+1/x)^2 = 5^2
x^2+2(x*1/x)+(1/x)^2 = 25
x^2+2+(1/x)^2 = 25
x^2+(1/x)^2 = 25-2
x^2+(1/x)^2 = 23
Then use the sum of cubes factoring formula.
a^3+b^3 = (a+b)(a^2-ab+b^2)
x^3+(1/x)^3 = (x+1/x)(x^2-x*(1/x)+(1/x)^2)
x^3+1/(x^3) = (x+1/x)(x^2-1+(1/x)^2)
x^3+1/(x^3) = (x+1/x)(x^2+(1/x)^2-1)
x^3+1/(x^3) = (5)(23-1)
x^3+1/(x^3) = (5)(22)
x^3+1/(x^3) = 110
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Another approach:
x + (1/x) = 5
x^2+1 = 5x ... multiply both sides by x to clear out the fraction
x^2-5x+1 = 0
Use the quadratic formula, or graphing calculator, to find these approximate solutions:
x = 0.2087 and x = 4.7913
Then you can plug either of those directly into x^3+(1/x)^3 to get 110 as the result. Your calculator may have some rounding error, so you might get very close to 110. The more decimal digits you use in the roots found above, the closer to 110 you should get.