If x=5+√3, what is x^3-1/x^3?

Answer 1

x=5+sqrt(3)

x^2 = 28 + 10*sqrt(3)

x^3 = (28 + 10*sqrt(3))(5 + sqrt(3))

= 140 + 78*sqrt(3) + 30

= 170 + 78*sqrt(3)

[(170 + 78*Sqrt(3))^2 - 1 ]/(170 + 78*sqrt(3))

Rationalizing that:

[(170 + 78*Sqrt(3))^2 - 1 ](170 - 78*sqrt(3))

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170^2 - 78^2*3

(28900 +26520*Sqrt(3)+18251)(170 - 78*Sqrt(3))

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28900 - 18252

[47151 + 26520*sqrt(3)](170 - 78*sqrt(3))

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10648

8015670 - 830622*sqrt(3) -6205680

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10648

1809990 - 830622*sqrt(3)

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10648