nthroot

Nth root of symbolic numbers

Syntax

y = nthroot(x,n)

Description

y = nthroot(x,n) returns the nth 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

Calculate Nth Roots

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 = $$1.0695539323639858023756790408254+0.2127475047267430357507130792184\hspace{0.17em}\mathrm{i}$$

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 = 
$$\left(\begin{array}{ccc}-3& -2& {\left(-1\right)}^{3/4}\hspace{0.17em}{4}^{1/4}\\ 3& 8& -\frac{\sqrt{12}\hspace{0.17em}\mathrm{i}}{12}\end{array}\right)$$

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 = 
$$\frac{\sqrt[n]{x}}{n\hspace{0.17em}x}$$

Input Arguments

x — Input array for taking root

symbolic array | numeric array

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)]

n — Input array for order of root

symbolic array | real array

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)

Version History

Introduced in R2018b