Rationalizing the Denominator

Imagine you have never seen a root sign, \sqrt {}, in your life.Consider the problem: Rationalize \frac{1}{5^\frac{1}{2}}.  This means that for some reason you want the denominator to have only integer exponents.  So \frac{1}{5^\frac{1}{2}}=\frac{1}{5^\frac{1}{2}}\frac{5^\frac{1}{2}}{5^\frac{1}{2}}=\frac{5^\frac{1}{2}}{5}.  If we start with Rationalize \frac{1}{5^\frac{2}{3}}, then \frac{1}{5^\frac{2}{3}}=\frac{1}{5^\frac{2}{3}}\frac{5^\frac{1}{3}}{5^\frac{1}{3}}=\frac{5^\frac{1}{3}}{5} and even if we get the problem, Rationalize \frac{1}{5^\frac{7}{9}}, then \frac{1}{5^\frac{7}{9}}=\frac{1}{5^\frac{7}{9}}\frac{5^\frac{2}{9}}{5^\frac{2}{9}}=\frac{5^\frac{2}{9}}{5}.

What if we were using roots? The above problems would be transformed as follows. Rationalize \frac{1}{\sqrt{5}}.  This means that for some reason you want the denominator to have no roots. So \frac{1}{\sqrt{5}}=\frac{1}{5^\frac{1}{2}}\frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{5}.  If we start with Rationalize \frac{1}{\sqrt[3]{5^2}}, then \frac{1}{\sqrt[3]{5^2}}=\frac{1}{\sqrt[3]{5^2}}\frac{\sqrt[3]{5}}{\sqrt[3]{5}}=\frac{\sqrt[3]{5}}{5} and even if we get the problem, Rationalize \frac{1}{\sqrt[9]{5^7}}, then \frac{1}{\sqrt[9]{5^7}}=\frac{1}{\sqrt[9]{5^7}}\frac{\sqrt[9]{5^2}}{\sqrt[9]{5^2}}=\frac{\sqrt[9]{5^2}}{5}

Which of the two techniques would be easier to explain?  Or to teach?  I think the first one.  This is one of the reasons I think that radical parts of an algebra curriculum can be dispensed with.  By the way the root method may not look to bad but that is only because I presented it second and your brain was already primed to see the pattern.

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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