The simplify/radical command is used to simplify expressions which contain radicals.  For the case of square roots only, see simplify/sqrt. Additional simplification facilities for expressions containing radicals include the radnormal, rationalize, and combine commands.

The simplify/radical command has three phases.  In the first phase, it tries to simplify each radical individually.  In the second phase, it looks at the set of all the radicals that appear in the expression and tries to eliminate radicals by writing one radical in terms of others.  Finally, the expression as a whole is normalized as a rational expression.

The main simplification applied to individual radicals is

Example 1: ${24}^{\frac{1}{3}}\to 23^{\frac{1}{3}}$.

Example 2: $\sqrt{{\mathrm{\pi }}^{2}x}\to \mathrm{\pi }\sqrt{x}$.

If sign of $x$ is unknown, the user can force Maple to do this transformation either by using the assume facility, or by specifying the symbolic option. If the sign of $x$ is unknown, and the symbolic option is given, $x$ is assumed real and positive.

The main simplification applied to the set of radicals is to try to write a radical as a constant times another radical.

Example 1: Given $\sqrt{30}+\sqrt{2}$, apply $\sqrt{30}\to \sqrt{15}\sqrt{2}$.

Example 2: Given $\sqrt{2x-2}+\sqrt{x-1}$, apply $\sqrt{2x-2}\to \sqrt{2}\sqrt{x-1}$.

The main simplification applied during the rationalization of the resulting expression is the following.

Given $x^{\frac{m}{n}}=x^{q+\frac{r}{n}}$ where $0<n$$<m$, write this as $x^q{A}^{\frac{r}{d}}$, normalize, and replace the dummy $A$ by $x$ on output.

Example:$\sqrt{{\left(x+1\right)}^3}-x\sqrt{x+1}\to \left(x+1\right)A-xA\to A\to \sqrt{x+1}$.

$e≔\left[{24}^{\frac{1}{3}}\&comma;{24}^{-\frac{1}{3}}\&comma;{\left(-24\right)}^{\frac{1}{3}}\right]$

$e≔\left[{24}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\&comma;\frac{{24}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{24}\&comma;{\left(\mathrm{-24}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]$

$\mathrm{simplify}\left(e\&comma;\mathrm{radical}\right)$

$\left[23^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\&comma;\frac{3^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{6}\&comma;\left(\mathrm{I}\sqrt{3}+1\right)3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]$

$e≔\left[6^{\frac{1}{3}}+{25}^{\frac{1}{3}}\&comma;6^{\frac{1}{3}}+{15}^{\frac{1}{3}}\right]$

$e≔\left[6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+{25}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\&comma;6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+{15}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]$

$\mathrm{simplify}\left(e\&comma;\mathrm{radical}\right)$

$\left[6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+5^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\&comma;3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\right]$

$e≔{\left(-8b^{3}a\right)}^{\frac{1}{3}}$

$e≔{\left(-8b^{3}a\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{simplify}\left(e\&comma;\mathrm{radical}\right)$

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

$\mathrm{simplify}\left(e\&comma;\mathrm{radical}\&comma;\mathrm{symbolic}\right)$

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

$\mathrm{assume}\left(b<0\right)$

$\mathrm{simplify}\left(e\&comma;\mathrm{radical}\right)$

$-2\mathrm{b~}{a}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$f≔{\left(x^4+3x^{3}y+3x^2{y}^2+x{y}^3\right)}^{\frac{1}{3}}$

$f≔{\left(x^4+3x^{3}y+3x^2{y}^2+x{y}^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{simplify}\left(f\&comma;\mathrm{radical}\right)$

${\left(x{\left(x+y\right)}^3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{simplify}\left(f\&comma;\mathrm{radical}\&comma;\mathrm{symbolic}\right)$

$\left(x+y\right)x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$f≔{\left(2x+2\right)}^{\frac{1}{3}}+{\left(4x+4\right)}^{\frac{1}{3}}+{\left(3x-3\right)}^{\frac{1}{3}}$

$f≔{\left(2x+2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+{\left(4x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+{\left(3x-3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{simplify}\left(f\&comma;\mathrm{radical}\right)$

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

$f≔{\left(x+1\right)}^{\frac{4}{3}}-x{\left(x+1\right)}^{\frac{1}{3}}$

$f≔{\left(x+1\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-x{\left(x+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{simplify}\left(f\&comma;\mathrm{radical}\right)$

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