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For a complex number x and integer n, surd(x, n) computes the nth root of x whose (complex) argument is closest to that of x. Ties are broken in such a way that the function x -> surd(x,n) is continuous in a counter-clockwise manner onto its branch cuts (that is, continuous in the direction of increasing complex argument).

In particular, if n is odd then if x>=0 then surd(x,n) = x^(1/n) and if x<0 then surd(x,n) = -(-x)^(1/n).

surd(-1, 3);

$\mathrm{-1}$

surd( 8, 3);

$2$

surd(-8, 3);

$\mathrm{-2}$

surd(-1, 2);

$\mathrm{I}$

surd(1+2*I,3);

${\left(1+2\mathrm{I}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

surd( x, n);

$\sqrt[n]{x}$

convert((6), power);

$x^{\frac{1}{n}}$

Maple simplifies the expression before converting. Constants will still be written with fractional exponents.

convert((9*x)^(1/3), surd);

$3^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt[3]{x}$

convert(3^(1/3)*x^(1/2)*a^b+f((-2)^(1/5)*x^(1/n)), surd);

$3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{x}{a}^b+f\left(-2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}{x}^{\frac{1}{n}}\right)$

int(surd(x^2,3)*(3*x^3-2*x^2+x-1), x=-2..2);

$-\frac{6122^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{55}$

Note the differences among the outputs of the surd, ^ , and root commands.

(8)^(1/3); root(8, 3); surd(8, 3);

$8^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$2$

$2$

(8.0)^(1/3); root(8.0, 3); surd(8.0, 3);

$2.000000000$

$2.000000000$

$2.000000000$

(-8)^(1/3); root(-8, 3); surd(-8, 3);

${\left(\mathrm{-8}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$2{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{-2}$

(-8.0)^(1/3); root(-8.0, 3); surd(-8.0, 3);

$1.000000000+1.732050807\mathrm{I}$

$1.000000000+1.732050807\mathrm{I}$

$\mathrm{-2.000000000}$