Rationalization is the transformation of a fractional expression's irrational denominator into a rational number. To do so, multiply the fractional expression by the appropriate whole fraction.
Multiply the numerator and denominator by the radical itself.
||\begin{align}\dfrac{2}{\sqrt{7}}&=\dfrac{2}{\sqrt{7}}\times\dfrac{\sqrt{7}}{\sqrt{7}}\\[3pt]&=\dfrac{2\times\sqrt{7}}{\sqrt{7}\times\sqrt{7}}\\[3pt]&=\dfrac{2\sqrt{7}}{7}\end{align}||
||\begin{align}\dfrac{3\sqrt{2}}{9\sqrt{22}}&=\dfrac{1}{3\sqrt{11}}\\[3pt]&=\dfrac{1\times\sqrt{11}}{3\sqrt{11}\times\sqrt{11}}\\[3pt]&=\dfrac{\sqrt{11}}{3\times11}\\[3pt]&=\dfrac{\sqrt{11}}{33}\end{align}||
||\begin{align}\dfrac{x+2}{\sqrt{2}}&=\dfrac{x+2}{\sqrt{2}}\times\dfrac{\sqrt{2}}{\sqrt{2}}\\[3pt]&=\dfrac{\sqrt{2}\times\left(x+2\right)}{\sqrt{2}\times\sqrt{2}}\\[3pt]&=\dfrac{\sqrt{2}\left(x\right)+2\sqrt{2}}{2}\end{align}||
First, identify the denominator’s conjugate, meaning the same expression but with the opposite operation. Then, multiply the numerator and the denominator by the conjugate.
||\begin{align}\dfrac{4}{3+\sqrt{2}}&=\dfrac{4}{3+\sqrt{2}}\times\dfrac{3-\sqrt{2}}{3-\sqrt{2}}\\[3pt]&=\dfrac{4\left(3-\sqrt{2}\right)}{\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)}\\[3pt]&=\dfrac{12-4\sqrt{2}}{\left(9-2\right)}\\[3pt]&=\dfrac{12-4\sqrt{2}}{7}\end{align}||When multiplying the two denominators, multiply the two binomials as follows:||\begin{align}\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right) &=\left(3\times3\right)+\left(3\times-\sqrt{2}\right)+\left(\sqrt{2}\times3\right)+\left(\sqrt{2}\times-\sqrt{2}\right)\\[3pt]&=9\boldsymbol{\color{#ec0000}{-3\sqrt{2}+3\sqrt{2}}}-2\\[3pt]&=9-2\\&=7\end{align}||