Radical

Author

Eric W. Weisstein
December 10, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Radical.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Radical.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Identities

((2^(1/3))+1) (((2^(1/3))-1)^(1/3))==(3^(1/3))

FullSimplify[(2^(1/3)+1)(2^(1/3)-1)^(1/3)]

(((2^(1/3))-1)^(1/3))==((1/9)^(1/3))-((2/9)^(1/3))+((4/9)^(1/3))

FullSimplify[((1/9)^(1/3)-(2/9)^(1/3)+(4/9)^(1/3))]

(Sqrt[(5^(1/3))-(4^(1/3))])==(1/3) ((2^(1/3))+(20^(1/3))-(25^(1/3)))

FullSimplify[1/3(2^(1/3)+20^(1/3)-25^(1/3))]

FullSimplify[(5^(1/3)-4^(1/3))^(1/2)]

FullSimplify[(5^(1/3)-4^(1/3))^(1/2)-1/3(2^(1/3)+20^(1/3)-25^(1/3))]

(cos^(1/3))(((2 π)/7))-((-cos(((4 π)/7)))^(1/3))-((-cos(((6 π)/7)))^(1/3))==-((((1/2) (-5+3 (7^(1/3)))))^(1/3))

Borwein and Bailey 2003, p. 77

CubeRoot[x_]:=Sign[x]Abs[x]^(1/3)

TraditionalForm[id1=Plus@@CubeRoot/@(Cos[#Pi/7]&/@{2,4,6})]

ToRadicals[RootReduce[Simplify[TrigToExp[id1]]]]

N/@{id1,-((((1/2) (-5+3 (7^(1/3)))))^(1/3))}

(cos^(1/3))(((2 π)/9))+(cos^(1/3))(((4 π)/9))-((-cos(((8 π)/9)))^(1/3)) ==(((-3+((3 (3^(2/3)))/2)))^(1/3))

TraditionalForm[id2=Plus@@CubeRoot/@(Cos[#Pi/9]&/@{2,4,8})]

ToRadicals[RootReduce[Simplify[TrigToExp[id2]]]]

N/@{id2,(((-3+((3 (3^(2/3)))/2)))^(1/3))}