Built-in Mathematica Symbol

ToRadicals

ToRadicals[expr]
attempts to express all Root objects in expr in terms of radicals.

ToRadicals can always give expressions in terms of radicals when the highest degree of the polynomial that appears in any Root object is four.

There are some cases in which expressions involving radicals can in principle be given, but ToRadicals cannot find them.

If Root objects in expr contain parameters, ToRadicals[expr] may yield a result that is not equal to expr for all values of the parameters.

ToRadicals automatically threads over lists, as well as equations, inequalities and logic functions.

All cubic Root objects can be converted into radicals:

All quartic Root objects can be converted into radicals:

Some higher-degree Root objects can be represented in terms of radicals:

ToRadicals also works with AlgebraicNumber objects:

ToRadicals converts trigonometric functions of rational multiples of

:

With Cubics->False the general formulas for solving cubic equations are not used:

Converting some cubic Root objects does not require the general formulas:

With Quartics->False the general formulas for solving quartic equations are not used:

Converting some quartic Root objects does not require the general formulas:

RootReduce[ToRadicals[r]]

r for any algebraic number r given as a Root object:

By default Reduce will not produce radical solutions for general cubics:

Use ToRadicals to convert:

Alternatively set Cubics->True:

In this case ToRadicals succeeds on the unreduced expression:

In this case ToRadicals succeeds on the reduced expression:

ToRadicals converts Root objects containing parameters:

The result may not be equal to the Root object for some values of the parameter: