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.

In cases where rationalization would lead to a 0 denominator, radnormal is first used to try to simplify the expression, which might result in a numeric exception. If radnormal has no effect, the input is returned unchanged.

$\mathrm{rationalize}\left(\frac{2}{2-\mathrm{sqrt}\left(2\right)}\right)$

$2+\sqrt{2}$

$\mathrm{rationalize}\left(\frac{1+2^{\frac{1}{3}}}{1-2^{\frac{1}{3}}}\right)$

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

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

$\frac{\left(1+\sqrt{x+y}\right)\left(x+y\right)}{-1+x+y}$

$\left[\frac{x}{x+\mathrm{sqrt}\left(3\right)}\,\frac{x}{x+\mathrm{sqrt}\left(1+\mathrm{sqrt}\left(3\right)\right)}\,\frac{x+y}{xy+\mathrm{sqrt}\left(3\right)+\mathrm{sqrt}\left(7\right)}\right]$

$\left[\frac{x}{x+\sqrt{3}}\,\frac{x}{x+\sqrt{1+\sqrt{3}}}\,\frac{x+y}{xy+\sqrt{3}+\sqrt{7}}\right]$

$\mathrm{rationalize}\left(\right)$

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

$\left[\frac{x+y}{x+\mathrm{sqrt}\left(y\right)}\,\frac{xy}{x+\mathrm{sqrt}\left(x+\mathrm{sqrt}\left(3\right)\right)}\right]$

$\left[\frac{x+y}{x+\sqrt{y}}\,\frac{xy}{x+\sqrt{x+\sqrt{3}}}\right]$

$\mathrm{rationalize}\left(\right)$

$\left[-\frac{\left(-x+\sqrt{y}\right)\left(x+y\right)}{x^2-y}\,-\frac{\left(-x+\sqrt{x+\sqrt{3}}\right)\left(x^2+\sqrt{3}-x\right)xy}{x^4-2x^3+x^2-3}\right]$

$\left[\frac{1}{\exp \left(\mathrm{sqrt}\left(x\right)\right)}\,\frac{x+y}{\cos \left(\mathrm{sqrt}\left(x\right)\right)+\sin \left(\mathrm{sqrt}\left(y\right)\right)}\right]$

$\left[\frac{1}{{\ⅇ}^{\sqrt{x}}}\,\frac{x+y}{\cos \left(\sqrt{x}\right)+\sin \left(\sqrt{y}\right)}\right]$

$\mathrm{rationalize}\left(\right)$

$\left[\frac{1}{{\ⅇ}^{\sqrt{x}}}\,\frac{x+y}{\cos \left(\sqrt{x}\right)+\sin \left(\sqrt{y}\right)}\right]$

$\mathrm{rationalize}\left(\frac{1}{1+\mathrm{root}\left(\sin \left(\frac{1}{1-\mathrm{sqrt}\left(\mathrm{\eta }\right)}\right)\,3\right)}\right)$

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

$\mathrm{rationalize}\left(\frac{1}{1+{\left(\frac{1}{1-\mathrm{sqrt}\left(a\right)}\right)}^{\frac{1}{3}}}\right)$

$\frac{\left({\left(-\frac{1}{-1+\sqrt{a}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-{\left(-\frac{1}{-1+\sqrt{a}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1\right)\left(-1+\sqrt{a}\right)\left(2+\sqrt{a}\right)}{-4+a}$

$\mathrm{rationalize}\left(\frac{1}{2+4^{\frac{1}{2}}}\right)$

$\frac{1}{4}$

$\mathrm{rationalize}\left(\frac{1}{2-4^{\frac{1}{2}}}\right)$

Error, (in rationalize) numeric exception: division by zero

$\mathrm{rationalize}\left(\frac{1}{x+{\left(x^2\right)}^{\frac{1}{2}}}\right)$

$\frac{1}{x+\sqrt{x^2}}$