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"Simplest radical form" as taught in secondary-school texts requires (1) no radicals in denominators, and (2) no denominators in radicals, and (3) only square-free numbers under square-root signs and similarly for higher indices.

As far as I know, the justification of this is that it is a way to tell whether expressions involving radicals are equal to each other. For example, one can thereby tell that $\dfrac{\sqrt5 - 1}{\sqrt5 + 2} = 3+\sqrt 5.$

Textbooks I have seen (and I have not seen most) command students dogmatically to put expressions into simplest radical form. I haven't seen them saying that the reason is to tell whether things are equal or not.

  • If one wishes the principal lesson to be "Here is a way to tell whether two expressions involving radicals are equal or not." then how should one design the exercises and the presentation of the idea?

  • Did I miss something where I said "As far as I know" above?

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    $\begingroup$ Another reason is to save work for the teacher, making it easier to tell if the final answer matches the answer key. $\endgroup$ – Robert Israel May 11 '17 at 16:07
  • $\begingroup$ @RobertIsrael : I don't see that that's another reason; to me that looks like the same reason. $\endgroup$ – Michael Hardy May 11 '17 at 16:09
  • $\begingroup$ It's not just the "canonical form" idea. It's also the idea that for numerical results (e.g., in the days before calculators), if one has to divide, one would prefer not to divide by a radical. Also, if one one must approximate a radical, one radical is better than two, and a smaller radicand is better than a bigger one. $\endgroup$ – quasi May 11 '17 at 16:09
  • $\begingroup$ Also, the dictum "always rationalize" is partly to force students to practice the method. The dictum is typically relaxed at the Calculus level, where it's common to see answers such as $1/\sqrt{2}$, rather than $\sqrt{2}/2$. $\endgroup$ – quasi May 11 '17 at 16:13
  • $\begingroup$ Ratonalizing the denominator (and sometimes the numerator) is needed for many Calculus limit calculations, so even though the dictum ("must rationalize") is relaxed, the students at the Calculus level are well aware (due to lots of practice at the precalculus level) that rationalization is at least an option. $\endgroup$ – quasi May 11 '17 at 16:21
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Aggregating my comments into an answer . . .

It's not just the "canonical form" idea. It's also the idea that for numerical results (e.g., in the days before calculators), if one has to divide, one would prefer not to divide by a radical. Also, if one one must approximate a radical expression, one radical is better than two, and a smaller radicand is better than a bigger one.

Even with a calculator, to get a numerical result for your example expression

$$\frac{\sqrt5 - 1}{\sqrt5 - 2}$$

I think the students would agree that it's easier (and less likely to be input incorrectly) to first rationalize by hand, and then use the calculator compute

$$3+\sqrt{5}$$

rather than using the calculator to evaluate the original expression.

As far as justifying the precalculus level dictum "always rationalize", I think it's partly to force students to practice the method. The dictum is typically relaxed at the Calculus level, where it's common to see answers such as ${\vphantom{{\displaystyle{\frac{x^2}{y^2}}}}}\large{\frac{1}{\sqrt{2}}}$ rather than $\large{\frac{\sqrt{2}}{2}}$.

Also, rationalizing the denominator (and sometimes the numerator) is needed for many Calculus limit calculations, so even though the dictum ("must rationalize") is relaxed, the students at the Calculus level are well aware (due to lots of practice at the precalculus level) that rationalization is at least an option.

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