Rationalizing Denominators

To rationalize a denominator, you first need to understand what is rational and irrational in terms of numbers.

Rational numbers: 3, 91, $$\sqrt{9}\,$$, 1

Irrational numbers: $$\pi\,$$, $$\sqrt{3}\,$$, 0.10100100010000...

The method to rationalizing the denominator:

If $$a < 0$$ and n is not a perfect square, then


 * $$\frac{a}{\sqrt{n}\,}\, * \frac{\sqrt{n}\,}{\sqrt{n}\,}\, = \frac{a\sqrt{n}\,}{n}\,$$.

This works, because


 * $$\frac{n}{n}\, = 1$$.

Therefore, multiplying by $$\frac{\sqrt{n}\,}{\sqrt{n}\,}\,$$ does not change the value of the fraction being multiplied. Hence,


 * $$\frac{a}{\sqrt{n}\,}\, = \frac{a\sqrt{n}\,}{n}\,$$.

Now the denominator is a rational number, and you can solve it.