Math: radicals: rationalizing the denominator

Tutoring math, you cover this topic with students in late middle school or early high school.  The math tutor shows the first case.

This article assumes that the reader is familiar with multiplying radicals. If necessary, see my article here about that.

Rationalizing the denominator is done when there’s a radical on the bottom of a fraction.  Consider the following:

 
Example 1: Simplify\ \frac{7}{2\sqrt{3}}

Solution:

To rationalize the denominator, you multiply the top and bottom by the same number. The number is chosen so it turns the radical on the bottom into a whole number:

(1)   \begin{equation*}\frac{7}{2\sqrt{3}}=\frac{7(\sqrt{3})}{2\sqrt{3}(\sqrt{3})}\end{equation*}

Above, we have multiplied \frac{7}{2\sqrt{3}}, top and bottom, by \sqrt{3}. We can do so because when you multiply the top and bottom by the same amount, the fraction’s value doesn’t change, just its form. Notice that \sqrt{3}(\sqrt{3})=\sqrt{9}=3. Therefore,

(2)   \begin{equation*}\frac{7(\sqrt{3})}{2\sqrt{3}(\sqrt{3})}=\frac{7\sqrt{3}}{{2(3)}}=\frac{7\sqrt{3}}{6}\end{equation*}

So it turns out that, when you rationalize the denominator, \frac{7}{2\sqrt{3}} becomes \frac{7\sqrt{3}}{6}.

Of course, there are more complicated situations where you have to rationalize the denominator. I’ll get to them in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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