The radnormal function performs normalization of expressions containing algebraic numbers in radical notation (see type/radnum), for example, $\sqrt{2}$ and ${\left(7+5\sqrt{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$. In particular, radnormal simplifies such a number to 0 if and only if it is mathematically equal to 0.

The expression $x^{\frac{1}{n}}$ is understood as the principal $n$th root of the object $x$ (see log).

Note that by default, if f is an algebraic function, then the denominator of f is not always rationalized.  To force rationalization of the denominator, use the option 'rationalized'.

For polynomials, radnormal attempts to preserve partial factorizations. Each factor is expanded and its coefficients are normalized. The option 'expanded' causes radnormal to expand all products.

Rational functions are expressed in the form a/b where a and b are normalized polynomials, and $\gcd \left(a\,b\right)=1$. Again, partial factorizations are preserved and the option 'expanded' applies.

The function radnormal is mapped over sets, lists, and relations.

Note that radnormal does handle expressions involving algebraic numbers in both indexed RootOf and radical notations (see RootOf/indexed).

The function radnormal sometimes unnests radicals, but does not always find an incidence of unnesting when it exists.

After some preliminary simplifications, a basis (in indexed RootOf notation) for the field generated by the radical numbers occurring in f is constructed and the numbers are expressed in this basis (see radfield). Then, the expression is normalized by evala@Normal and the numbers are converted back to radical notation.

If infolevel[radnormal] is assigned a positive integer, then information about the execution of the program will be displayed.

$a≔\mathrm{sqrt}\left(2\right)\mathrm{sqrt}\left(3\right)-\mathrm{sqrt}\left(6\right)$

$a≔\sqrt{2}\sqrt{3}-\sqrt{6}$

$\mathrm{radnormal}\left(a\right)$

$0$

$a≔{\left(7+5\mathrm{sqrt}\left(2\right)\right)}^{\frac{1}{3}}$

$a≔{\left(7+5\sqrt{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{radnormal}\left(a\right)$

$1+\sqrt{2}$

$a≔\frac{1}{2^{\frac{1}{2}}+3^{\frac{1}{2}}+6^{\frac{1}{2}}}$

$a≔\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}$

$\mathrm{radnormal}\left(a\right)$

$\frac{5\sqrt{3}}{23}-\frac{\sqrt{2}\sqrt{3}}{23}-\frac{12}{23}+\frac{7\sqrt{2}}{23}$

$a≔\frac{2^{\frac{1}{4}}\left(2^{\frac{1}{2}}+2\right)}{{\left(8+62^{\frac{1}{2}}\right)}^{\frac{1}{2}}}$

$a≔\frac{2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(2+\sqrt{2}\right)}{\sqrt{8+6\sqrt{2}}}$

$\mathrm{radnormal}\left(a\right)$

$1$

$a≔\frac{x^2+2x{2}^{\frac{1}{2}}-2^{\frac{1}{2}}x{6}^{\frac{1}{2}}+5-22^{\frac{1}{2}}{3}^{\frac{1}{2}}}{x^2-2x{3}^{\frac{1}{2}}+1}$

$a≔\frac{x^2+2x\sqrt{2}-\sqrt{2}x\sqrt{6}+5-2\sqrt{2}\sqrt{3}}{x^2-2x\sqrt{3}+1}$

$\mathrm{radnormal}\left(a\right)$

$\frac{x+\sqrt{2}-\sqrt{3}}{x-\sqrt{3}-\sqrt{2}}$

$a≔\left(x-6^{\frac{1}{2}}\right)\left(x-2^{\frac{1}{2}}{3}^{\frac{1}{2}}\right)$

$a≔\left(x-\sqrt{6}\right)\left(x-\sqrt{2}\sqrt{3}\right)$

$\mathrm{radnormal}\left(a\right)$

${\left(x-\sqrt{2}\sqrt{3}\right)}^2$

$\mathrm{radnormal}\left(a\,'\mathrm{expanded}'\right)$

$6-2\sqrt{2}\sqrt{3}x+x^2$

$a≔\frac{1}{\mathrm{sqrt}\left(x\right)+\mathrm{sqrt}\left(y\right)-\mathrm{sqrt}\left(z\right)}$

$a≔\frac{1}{\sqrt{x}+\sqrt{y}-\sqrt{z}}$

$\mathrm{radnormal}\left(a\right)$

$\frac{1}{\sqrt{x}+\sqrt{y}-\sqrt{z}}$

$\mathrm{radnormal}\left(a\,'\mathrm{rationalized}'\right)$

$-\frac{2\sqrt{x}\sqrt{z}\sqrt{y}-x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\sqrt{x}y+\sqrt{x}z+x\sqrt{y}-y^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+z\sqrt{y}-\sqrt{z}x-\sqrt{z}y+z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^2-2xy-2xz+y^2-2yz+z^2}$

$a≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+1\,\mathrm{index}=1\right)-{\left(-1\right)}^{\frac{2}{3}}$

$a≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+1\,\mathrm{index}=1\right)-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{radnormal}\left(a\right)$

$0$