 rationalize - Maple Help

rationalize

rationalize denominator Calling Sequence rationalize(expr) Parameters Description

 • The function rationalize attempts to rationalize the given expression, removing all roots from the denominator.
 • rationalize does not operate inside transcendental functions, such as exp and sin. Radical expressions that are arguments to such functions within expr will not be rationalized. Examples

 > $\mathrm{rationalize}\left(\frac{2}{2-\mathrm{sqrt}\left(2\right)}\right)$
 ${2}{+}\sqrt{{2}}$ (1)
 > $\mathrm{rationalize}\left(\frac{1+{2}^{\frac{1}{3}}}{1-{2}^{\frac{1}{3}}}\right)$
 ${-}\left({1}{+}{{2}}^{{1}}{{3}}}\right){}\left({{2}}^{{2}}{{3}}}{+}{{2}}^{{1}}{{3}}}{+}{1}\right)$ (2)
 > $\mathrm{rationalize}\left(\frac{x+y}{\mathrm{sqrt}\left(x+y\right)-1}\right)$
 $\frac{\left({1}{+}\sqrt{{x}{+}{y}}\right){}\left({x}{+}{y}\right)}{{-}{1}{+}{x}{+}{y}}$ (3)
 > $\left[\frac{x}{x+\mathrm{sqrt}\left(3\right)},\frac{x}{x+\mathrm{sqrt}\left(1+\mathrm{sqrt}\left(3\right)\right)},\frac{x+y}{xy+\mathrm{sqrt}\left(3\right)+\mathrm{sqrt}\left(7\right)}\right]$
 $\left[\frac{{x}}{{x}{+}\sqrt{{3}}}{,}\frac{{x}}{{x}{+}\sqrt{{1}{+}\sqrt{{3}}}}{,}\frac{{x}{+}{y}}{{x}{}{y}{+}\sqrt{{3}}{+}\sqrt{{7}}}\right]$ (4)
 > $\mathrm{rationalize}\left(\right)$
 $\left[{-}\frac{\left({-}{x}{+}\sqrt{{3}}\right){}{x}}{{{x}}^{{2}}{-}{3}}{,}{-}\frac{\left({-}{x}{+}\sqrt{{1}{+}\sqrt{{3}}}\right){}\left({{x}}^{{2}}{+}\sqrt{{3}}{-}{1}\right){}{x}}{{{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{-}{2}}{,}\frac{\left({-}{x}{}{y}{+}\sqrt{{7}}{-}\sqrt{{3}}\right){}\left({-}{{x}}^{{2}}{}{{y}}^{{2}}{+}{2}{}\sqrt{{3}}{}{x}{}{y}{+}{4}\right){}\left({x}{+}{y}\right)}{{{x}}^{{4}}{}{{y}}^{{4}}{-}{20}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{16}}\right]$ (5)
 > $\left[\frac{x+y}{x+\mathrm{sqrt}\left(y\right)},\frac{xy}{x+\mathrm{sqrt}\left(x+\mathrm{sqrt}\left(3\right)\right)}\right]$
 $\left[\frac{{x}{+}{y}}{{x}{+}\sqrt{{y}}}{,}\frac{{x}{}{y}}{{x}{+}\sqrt{{x}{+}\sqrt{{3}}}}\right]$ (6)
 > $\mathrm{rationalize}\left(\right)$
 $\left[{-}\frac{\left({-}{x}{+}\sqrt{{y}}\right){}\left({x}{+}{y}\right)}{{{x}}^{{2}}{-}{y}}{,}{-}\frac{\left({-}{x}{+}\sqrt{{x}{+}\sqrt{{3}}}\right){}\left({{x}}^{{2}}{+}\sqrt{{3}}{-}{x}\right){}{x}{}{y}}{{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{+}{{x}}^{{2}}{-}{3}}\right]$ (7)

Radical expressions that are arguments to transcendental functions, such as exp and sin will not be rationalized.

 > $\left[\frac{1}{\mathrm{exp}\left(\mathrm{sqrt}\left(x\right)\right)},\frac{x+y}{\mathrm{cos}\left(\mathrm{sqrt}\left(x\right)\right)+\mathrm{sin}\left(\mathrm{sqrt}\left(y\right)\right)}\right]$
 $\left[\frac{{1}}{{{ⅇ}}^{\sqrt{{x}}}}{,}\frac{{x}{+}{y}}{{\mathrm{cos}}{}\left(\sqrt{{x}}\right){+}{\mathrm{sin}}{}\left(\sqrt{{y}}\right)}\right]$ (8)
 > $\mathrm{rationalize}\left(\right)$
 $\left[\frac{{1}}{{{ⅇ}}^{\sqrt{{x}}}}{,}\frac{{x}{+}{y}}{{\mathrm{cos}}{}\left(\sqrt{{x}}\right){+}{\mathrm{sin}}{}\left(\sqrt{{y}}\right)}\right]$ (9)
 > $\mathrm{rationalize}\left(\frac{1}{1+\mathrm{root}\left(\mathrm{sin}\left(\frac{1}{1-\mathrm{sqrt}\left(\mathrm{\eta }\right)}\right),3\right)}\right)$
 $\frac{{{\mathrm{sin}}{}\left(\frac{{1}}{{1}{-}\sqrt{{\mathrm{\eta }}}}\right)}^{{2}}{{3}}}{-}{{\mathrm{sin}}{}\left(\frac{{1}}{{1}{-}\sqrt{{\mathrm{\eta }}}}\right)}^{{1}}{{3}}}{+}{1}}{{1}{+}{\mathrm{sin}}{}\left(\frac{{1}}{{1}{-}\sqrt{{\mathrm{\eta }}}}\right)}$ (10)
 > $\mathrm{rationalize}\left(\frac{1}{1+{\left(\frac{1}{1-\mathrm{sqrt}\left(a\right)}\right)}^{\frac{1}{3}}}\right)$
 $\frac{\left({\left({-}\frac{{1}}{{-}{1}{+}\sqrt{{a}}}\right)}^{{2}}{{3}}}{-}{\left({-}\frac{{1}}{{-}{1}{+}\sqrt{{a}}}\right)}^{{1}}{{3}}}{+}{1}\right){}\left({-}{1}{+}\sqrt{{a}}\right){}\left({2}{+}\sqrt{{a}}\right)}{{-}{4}{+}{a}}$ (11)