Rationalizing

A fraction is not considered simplified until all the radicals have been removed from the denominator. If a denominator contains a single term with a square root, it can be rationalized by multiplying both the numerator and denominator by that square root. If the denominator contains square roots separated by a plus or minus sign, then multiply both the numerator and denominator by the conjugate, which is formed by merely changing the sign between the roots.

Example 1: Rationalize the fraction $\frac{2}{3\sqrt{5}}$

Multiply top and bottom of the fraction by âˆš5

$\frac{2}{3\sqrt{5}}Â·\frac{\sqrt{5}}{\sqrt{5}}$ = $\frac{2\sqrt{5}}{3Â·\sqrt{25}}$ = $\frac{2\sqrt{5}}{3Â·5}$ = $\frac{2\sqrt{5}}{15}$

Example 2: Rationalize the fraction $\frac{2}{3-\sqrt{5}}$

Multiply top and bottom of the fraction by the conjugate (3+âˆš5):

$\frac{2}{3-\sqrt{5}}Â·\frac{3+\sqrt{5}}{3+\sqrt{5}}$ = $\frac{2\left(3+\sqrt{5}\right)}{{3}^{2}+3\sqrt{5}-3\sqrt{5}-{\left(\sqrt{5}\right)}^{2}}$

= $\frac{2\left(3+\sqrt{5}\right)}{9-5}$ = $\frac{2\left(3+\sqrt{5}\right)}{4}$

= $\frac{3+\sqrt{5}}{2}$