Basic Rationalizing of Denominators
Mini Video Lecture
Key Takeaways:
Occasionally, when radicals appear in the denominator of a fraction, it is convenient to rewrite the fraction in a way that gets the radical out of the denominator.
The process by which we do this is called “Rationalizing the Denominator.”
For example: Suppose we have the expression \(\frac{1}{\sqrt{x}}\) and we don’t want the radical in the denominator. To get rid of it, we can cleverly multiply the expression by \(1\), or rather an expression that is equal to \(1\) so that we don’t change the value of the expression. Viz:
$$\frac{1}{\sqrt{x}}=\frac{1}{\sqrt{x}}\cdot \frac{\sqrt{x}}{\sqrt{x}}=\frac{\sqrt{x}}{\sqrt{x^2}}=\frac{\sqrt{x}}{|x|}$$
Notice that multiplying the numerator and denominator of the original fraction by \(\sqrt{x}\) effectively squares what is already in the denominator, cancelling the radical. Overall, the idea is to multiply the numerator and denominator by whatever gets rid of the radical on the denominator.
Hence, when rationalizing the denominator when you have only a radical in said denominator, just multiply the top and bottom of the fraction by that same radical and simplify. Like magic, your denominator radical vanishes.
NOTE: You may have heard a teacher say that “you can never have a radical in the denominator!” This is false and there is nothing inherently wrong/incorrect about having a radical in the denominator. In fact, it is occasionally more useful to keep radicals down there! More often though, for more complicated problems such those in calculus, it is helpful to rationalize denominators.