# Documentation

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# simplifyRadical

Simplify radicals in arithmetical expressions

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## Syntax

```simplifyRadical(`z`)
```

## Description

`simplifyRadical(z)` tries to simplify the radicals in the expression `z`. The result is mathematically equivalent to `z`.

`radsimp` and `simplifyRadical` are equivalent.

## Examples

### Example 1

Simplify these constant expressions with square roots and higher order radicals:

```simplifyRadical(3*sqrt(7)/(sqrt(7) - 2)), simplifyRadical(sqrt(5 + 2*sqrt(6))); simplifyRadical(sqrt(5*sqrt(3) + 6*sqrt(2))), simplifyRadical(sqrt(3 + 2*sqrt(2)))```
``` ```
``` ```
`simplifyRadical((1/2 + 1/4*3^(1/2))^(1/2))`
``` ```
`simplifyRadical((5^(1/3) - 4^(1/3))^(1/2))`
``` ```
```simplifyRadical(sqrt(3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2)))) + 14))```
``` ```
`simplifyRadical(2*2^(1/4) + 2^(3/4) - (6*2^(1/2) + 8)^(1/2))`
``` ```
```simplifyRadical(sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3)))```
``` ```

### Example 2

Create the following expression and then simplify it using `simplifyRadical`:

`x := sqrt(3)*I/2 + 1/2: y := x^(1/3) + x^(-1/3): z := y^3 - 3*y`
``` ```
`simplifyRadical(z)`
``` ```
`delete x, y, z:`

### Example 3

Use `simplifyRadical` to simplify these arithmetical expressions containing variables:

`z := x/(sqrt(3) - 1) - x/2`
``` ```
`simplifyRadical(z) = expand(radsimp(z))`
``` ```
`delete z:`

### Example 4

Use `simplifyRadical` to simplify nested radicals. When simplifying nested radicals, `simplifyRadical` tries to reduce the nesting depth:

```simplifyRadical((6*2^(1/2) + 8)^(1/2)); simplifyRadical(((32/5)^(1/5) - (27/5)^(1/5))^(1/3)); simplifyRadical(sqrt((3+2^(1/3))^(1/2) * (4-2^(1/3))^(1/2)))```
``` ```
``` ```
``` ```

## Parameters

 `z`

## Return Values

Arithmetical expression.

## Algorithms

For constant algebraic expressions, `simplifyRadical` constructs a tower of algebraic extensions of using the domain `Dom::AlgebraicExtension`. It tries to return the simplest possible form.

This function is based on an algorithm described in Borodin, Fagin, Hopcroft and Tompa, "Decreasing the Nesting Depth of Expressions Involving Square Roots", JSC 1, 1985, pp. 169-188.In some special cases, an algorithm based on Landau, "How to tangle with a nested radical", The Mathematical Intelligencer 16, 1994, no. 2, pp. 49-55, is used.

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