normalize an expression containing radical numbers

Parameters

 f - algebraic expression opts1, opts2, ... - sequence of options

Description

 • The radnormal function performs normalization of expressions containing algebraic numbers in radical notation (see type/radnum), for example, $\sqrt{2}$ and ${\left(7+5\sqrt{2}\right)}^{1/3}$. 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 a number, 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 $\mathrm{gcd}\left(a,b\right)=1$. 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 will sometimes unnest radicals, but will 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.

Examples

 > $a≔\sqrt{2}\sqrt{3}-\sqrt{6}$
 ${a}{≔}\sqrt{{2}}{}\sqrt{{3}}{-}\sqrt{{6}}$ (1)
 > $\mathrm{radnormal}\left(a\right)$
 ${0}$ (2)
 > $a≔{\left(7+5\sqrt{2}\right)}^{\frac{1}{3}}$
 ${a}{≔}{\left({7}{+}{5}{}\sqrt{{2}}\right)}^{{1}{/}{3}}$ (3)
 > $\mathrm{radnormal}\left(a\right)$
 $\sqrt{{2}}{+}{1}$ (4)
 > $a≔\frac{1}{{2}^{\frac{1}{2}}+{3}^{\frac{1}{2}}+{6}^{\frac{1}{2}}}$
 ${a}{≔}\frac{{1}}{\sqrt{{2}}{+}\sqrt{{3}}{+}\sqrt{{6}}}$ (5)
 > $\mathrm{radnormal}\left(a\right)$
 $\frac{{1}}{\sqrt{{2}}{+}\sqrt{{3}}{+}\sqrt{{2}}{}\sqrt{{3}}}$ (6)
 > $\mathrm{radnormal}\left(a,'\mathrm{rationalized}'\right)$
 $\frac{{5}}{{23}}{}\sqrt{{3}}{-}\frac{{1}}{{23}}{}\sqrt{{2}}{}\sqrt{{3}}{-}\frac{{12}}{{23}}{+}\frac{{7}}{{23}}{}\sqrt{{2}}$ (7)
 > $a≔\frac{{2}^{\frac{1}{4}}\left({2}^{\frac{1}{2}}+2\right)}{{\left(8+6{2}^{\frac{1}{2}}\right)}^{\frac{1}{2}}}$
 ${a}{≔}\frac{{{2}}^{{1}{/}{4}}{}\left(\sqrt{{2}}{+}{2}\right)}{\sqrt{{8}{+}{6}{}\sqrt{{2}}}}$ (8)
 > $\mathrm{radnormal}\left(a\right)$
 ${1}$ (9)
 > $a≔\frac{{x}^{2}+2x{2}^{\frac{1}{2}}-{2}^{\frac{1}{2}}x{6}^{\frac{1}{2}}+5-2{2}^{\frac{1}{2}}{3}^{\frac{1}{2}}}{{x}^{2}-2x{3}^{\frac{1}{2}}+1}$
 ${a}{≔}\frac{{{x}}^{{2}}{+}{2}{}{x}{}\sqrt{{2}}{-}\sqrt{{2}}{}{x}{}\sqrt{{6}}{+}{5}{-}{2}{}\sqrt{{2}}{}\sqrt{{3}}}{{{x}}^{{2}}{-}{2}{}{x}{}\sqrt{{3}}{+}{1}}$ (10)
 > $\mathrm{radnormal}\left(a\right)$
 $\frac{{-}\sqrt{{3}}{+}\sqrt{{2}}{+}{x}}{{x}{-}\sqrt{{3}}{-}\sqrt{{2}}}$ (11)
 > $a≔\left(x-{6}^{\frac{1}{2}}\right)\left(x-{2}^{\frac{1}{2}}{3}^{\frac{1}{2}}\right)$
 ${a}{≔}\left({x}{-}\sqrt{{6}}\right){}\left({x}{-}\sqrt{{2}}{}\sqrt{{3}}\right)$ (12)
 > $\mathrm{radnormal}\left(a\right)$
 ${\left({x}{-}\sqrt{{2}}{}\sqrt{{3}}\right)}^{{2}}$ (13)
 > $\mathrm{radnormal}\left(a,'\mathrm{expanded}'\right)$
 ${6}{-}{2}{}\sqrt{{3}}{}\sqrt{{2}}{}{x}{+}{{x}}^{{2}}$ (14)
 > $a≔\frac{1}{\sqrt{x}+\sqrt{y}-\sqrt{z}}$
 ${a}{≔}\frac{{1}}{\sqrt{{x}}{+}\sqrt{{y}}{-}\sqrt{{z}}}$ (15)
 > $\mathrm{radnormal}\left(a\right)$
 $\frac{{1}}{\sqrt{{x}}{+}\sqrt{{y}}{-}\sqrt{{z}}}$ (16)
 > $\mathrm{radnormal}\left(a,'\mathrm{rationalized}'\right)$
 ${-}\frac{{2}{}\sqrt{{x}}{}\sqrt{{z}}{}\sqrt{{y}}{-}{{x}}^{{3}{/}{2}}{+}\sqrt{{x}}{}{y}{+}\sqrt{{x}}{}{z}{+}{x}{}\sqrt{{y}}{-}{{y}}^{{3}{/}{2}}{+}{z}{}\sqrt{{y}}{-}\sqrt{{z}}{}{x}{-}\sqrt{{z}}{}{y}{+}{{z}}^{{3}{/}{2}}}{{{x}}^{{2}}{-}{2}{}{x}{}{y}{-}{2}{}{x}{}{z}{+}{{y}}^{{2}}{-}{2}{}{y}{}{z}{+}{{z}}^{{2}}}$ (17)
 > $a≔\mathrm{RootOf}\left({\mathrm{_Z}}^{2}+\mathrm{_Z}+1,\mathrm{index}=1\right)-{\left(-1\right)}^{\frac{2}{3}}$
 ${a}{≔}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{\mathrm{_Z}}{+}{1}{,}{\mathrm{index}}{=}{1}\right){-}{\left({-}{1}\right)}^{{2}{/}{3}}$ (18)
 > $\mathrm{radnormal}\left(a\right)$
 ${0}$ (19)