Fractions Involving Radicals

In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator. When we are given expressions that involve radicals in the denominator, it makes it easier to evaluate the expression if we rewrite it in a way that the radical is no longer in the denominator. This process is called rationalizing the denominator.

Before we begin, remember that whatever we do to one side of an algebraic equation, we must also do to the other side. This same principal can be applied to fractions: whatever we do to the numerator, we must also do to the denominator, and vice versa.  

Let's look at an example to illustrate the process of rationalizing the denominator.

You are given the fraction $\frac{10}{\sqrt{3}}$, and you want to simplify it by eliminating the radical from the denominator. Recall that a radical multiplied by itself equals its radicand, or the value under the radical sign. Therefore, multiply the top and bottom of the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, and watch how the radical expression disappears from the denominator:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$