ToRadicals

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`ToRadicals`

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ToRadicals[expr]

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

Details and Options

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.

Examples

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Basic Examples (1)Summary of the most common use cases

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-hawleo

Out[1]=1

Scope (3)Survey of the scope of standard use cases

All cubic Root objects can be converted into radicals:

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-zv765

Out[1]=1

All quartic Root objects can be converted into radicals:

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-dowe7k

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-i5uzsz

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-ms22m8

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0b8c1pqrzy-drcsim

Out[3]=3

ToRadicals also works with AlgebraicNumber objects:

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-fqsdp4

Out[1]=1

Generalizations & Extensions (1)Generalized and extended use cases

ToRadicals converts trigonometric functions of rational multiples of :

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-eqquo6

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-cr16f5

Out[2]=2

Options (4)Common values & functionality for each option

Cubics (2)

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

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-nkk65p

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-ln2for

Out[1]=1

Quartics (2)

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

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-jv5j2s

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-bhvu5z

Out[1]=1

Properties & Relations (2)Properties of the function, and connections to other functions

RootReduce[ToRadicals[r]]==r for any algebraic number r given as a Root object:

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-jku8c7

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-bp9chv

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-grqg30

Out[1]=1

Use ToRadicals to convert:

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-b4ze1o

Out[2]=2

Alternatively set Cubics->True:

In[3]:=3

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https://wolfram.com/xid/0b8c1pqrzy-lkitdf

Out[3]=3

Possible Issues (3)Common pitfalls and unexpected behavior

In this case ToRadicals succeeds on the unreduced expression:

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-tc3v7

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-kj87yy

Out[2]=2

In this case ToRadicals succeeds on the reduced expression:

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-ge96cd

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-gtvshj

Out[2]=2

ToRadicals converts Root objects containing parameters:

In[1]:=1

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https://wolfram.com/xid/0b8c1pqrzy-gp3jpj

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0b8c1pqrzy-vnvc

Out[2]=2

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