# nthroot

Nth root of symbolic numbers

## Syntax

``y = nthroot(x,n)``

## Description

example

````y = nthroot(x,n)` returns the `n`th root of `x` with the phase angle closest to the phase of `x`. The output `y` has symbolic data type if any input argument is symbolic. The variables satisfy `y.^n = x`.```

## Examples

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Calculate the nth root of a negative number.

```x = sym(-27); n = -3; y = nthroot(x,n)```
```y =  $-\frac{1}{3}$```

Check that the answer solves the equation ${y}^{n}=x$.

`y^n`
`ans = $-27$`

Calculate the nth root of a complex number.

```x = sym(1 + 1i); y = nthroot(x,4)```
`y = ${\left(1+\mathrm{i}\right)}^{1/4}$`

Find a numeric equivalent of the root.

`vpa(y)`
`ans = $1.0695539323639858023756790408254+0.2127475047267430357507130792184 \mathrm{i}$`

Check that the answer solves the equation ${y}^{n}=x$.

`y^4`
`ans = $1+\mathrm{i}$`

Calculate the nth roots of an array.

```x = sym([-27,-8,-4 27,64,-12])```
```x =  $\left(\begin{array}{ccc}-27& -8& -4\\ 27& 64& -12\end{array}\right)$```
```n = sym([3,3,4 3,2,-2])```
```n =  $\left(\begin{array}{ccc}3& 3& 4\\ 3& 2& -2\end{array}\right)$```
`y = nthroot(x,n)`
```y =  $\left(\begin{array}{ccc}-3& -2& {\left(-1\right)}^{3/4} {4}^{1/4}\\ 3& 8& -\frac{\sqrt{12} \mathrm{i}}{12}\end{array}\right)$```

Check that the answer solves the equation ${y}^{n}=x$.

`y.^n`
```ans =  $\left(\begin{array}{ccc}-27& -8& -4\\ 27& 64& -12\end{array}\right)$```

Use `nthroot` in further symbolic calculations.

```syms x y = solve(nthroot(x,-3) == -3, x)```
```y =  $-\frac{1}{27}$```
```syms x n y = diff(nthroot(x,n),x)```
```y =  $\frac{\sqrt[n]{x}}{n x}$```

## Input Arguments

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Input array for taking root, specified as a symbolic or numeric array. When taking the root, the function acts element-wise.

If both `x` and `n` are nonscalar arrays, they must have the same size. If any element of `x` or `n` is symbolic and some elements are numeric, `nthroot` converts numeric arguments to symbolic before processing.

Example: `[sym(-8),sym(8);sym(-27),sym(27)]`

Input array for order of root, specified as a symbolic array or real array.

• If an element of `x` is not real and positive, meaning it is either negative or has a nonzero imaginary part, then the corresponding element of `n` must be a nonzero integer.

• If an element of `x` is real and positive, then the corresponding element of `n` can have any nonzero real value.

If both `x` and `n` are nonscalar arrays, they must have the same size. If any element of `x` or `n` are symbolic and some elements are numeric, `nthroot` converts numeric arguments to symbolic before processing.

Example: `sym(-3)`