## Radicals and I Don’t Get Along.

We have reached the point in the term when we are studying radical expressions.  The author of our text to his credit defines roots in the first section of the chapter and introduces the equivalent rational exponent notation in the second.  Then he spends the next three sections in manipulating radical expressions.  I fell into his trap again.

Case in point.  Consider this problem: Simplify $\sqrt{x^17}$.  Immediately we have a problem.  This looks pretty simple to me already.  So does $\sqrt{20}$ versus $2\sqrt{5}$ and $\sqrt{20}$ is easier to enter into the calculator.  Back to $\sqrt{x^17}$.  The author actually says “simplify by factoring” and expects us to get $x^8\sqrt{x}$ as the simplification.  Note that he, like most authors, assumes in this section that $x\ge 0$.

I take it that this exercise exists not to prepare students for future encounters with like expressions but to get them used to the rule $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$. But we already know this as $(ab)^\frac{1}{n}=(a)^\frac{1}{n}(b)^\frac{1}{n}$.  If the object is to just get  $x^8\sqrt{x}$ then I  should have taught the following  sequence of steps.  Change $\sqrt{x^17}$ to  $x^\frac{17}{2}$.  Change $\frac{17}{2}$ to a proper fraction $8\frac{1}{2}$.  (Note:  I tried this in my next class and most students did not know how to change an improper fraction to a proper fraction and there is no single command on the TI graphing calculators that does this.)  Then $\sqrt{x^17} = x^8x^\frac{1}{2} = x^8\sqrt{x}$.  This method is straightforward though it takes a little explaining and generalizes to this type of problem: Simplify $\sqrt[7]{x^{123}}$.  Actually the author was willing to do this for $\sqrt[4]{x^{20}}$ where things come out even.

Radicals and I don’t get along and I have to watch myself or I just start going through the motions when I get to these sections.