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The even-even-odd rule is a mnemonic that is helpful for students for simplifying radical expressions. The phrase even-even-odd stands for the rule: If a real variable to an even exponent is under a radical with an even index and, when the radical is eliminated, the resulting exponent on the variable is odd, then absolute value signs must be placed around the variable. (All numbers to which "exponent" and "index" refer are natural numbers.) This rule is justified by the following:
Recall that, for any positive integer  ,  is the  th root of  if and only if  and
 . Thus, for any positive integer  and
 ,
The following are some examples of how to use the even-even-odd rule.
Problem Let  , and  be real variables. Simplify the expression
![$ \sqrt[4]{x^{12}y^8}$](http://images.planetmath.org:8080/cache/objects/8054/l2h/img12.png "$ \sqrt[4]{x^{12}y^8}$") .
Solution: The exponent on the  is even (12), the index of the radical is even (4), and the exponent that will occur on the  once the radical is eliminated will be odd (3). Thus, absolute values are necessary on the  .
The exponent on the  is even (8), the index of the radical is even (4), and the exponent that will occur on the  once the radical is eliminated will be even (2). Thus, according to the rule, absolute values are not necessary on the  . (Note, though, that it would not be incorrect to have them.) The reason that the absolute values are not necessary is that  is nonnegative regardless of the value of  .
Thus, we have
![$ \sqrt[4]{x^{12}y^8}=\vert x\vert^3y^2$](http://images.planetmath.org:8080/cache/objects/8054/l2h/img21.png "$ \sqrt[4]{x^{12}y^8}=\vert x\vert^3y^2$") . (The answer  is also acceptable.)
Some care is needed in applying the even-even-odd rule, as the next problem shows.
Problem Let  be a real variable. Simplify the expression
![$ \sqrt[4]{x^2}$](http://images.planetmath.org:8080/cache/objects/8054/l2h/img24.png "$ \sqrt[4]{x^2}$") .
Note that, as stated, the even-even-odd rule does not apply here, since, if the radical were eliminated, the resulting exponent on the  will be
 . On the other hand, it can still be used to provide a correct answer for this particular problem.
Solution:
The good news is that, for square roots, this issue discussed above does not arise: If the even-even-odd rule does not apply, then absolute values are not necessary. That is because, if
 is odd, the expression
 only makes sense in the real numbers when  is nonnegative.
I would like to thank Mrs. Sue Millikin, who taught me how to simplify radical expressions in this manner.
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![$\displaystyle \sqrt[n]{x^n}=\begin{cases} \vert x\vert & \text{if } n \text{ is even} \ x & \text{if } n \text{ is odd.} \end{cases}$](http://images.planetmath.org:8080/cache/objects/8054/l2h/img9.png "$\displaystyle \sqrt[n]{x^n}=\begin{cases} \vert x\vert & \text{if } n \text{ is even} \ x & \text{if } n \text{ is odd.} \end{cases}$")
![$\displaystyle \sqrt[4]{x^2}=\sqrt{\sqrt{x^2}}=\sqrt{\vert x\vert}$](http://images.planetmath.org:8080/cache/objects/8054/l2h/img27.png "$\displaystyle \sqrt[4]{x^2}=\sqrt{\sqrt{x^2}}=\sqrt{\vert x\vert}$")