even-even-odd rule

The 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 (http://planetmath.org/Exponent2) is under a with an even index (http://planetmath.org/Radical6) and, when the is eliminated, the resulting on the variableMathworldPlanetmath is odd, then absolute valueMathworldPlanetmathPlanetmathPlanetmath signs must be placed around the variable. (All numbers to which ”” and ”” refer are natural numbers.) This rule is justified by the following:

Recall that, for any positive integer n, b is the nth root (http://planetmath.org/NthRoot) of a if and only if bn=a and sign(b)=sign(a). Thus, for any positive integer n and x,

xnn={|x|if n is evenxif n is odd.

The following are some examples of how to use the even-even-odd rule.


Let x, and y be real variables. Simplify the expression x12y84.

Solution:  The on the x is even (12), the of the is even (4), and the that will occur on the x once the is eliminated will be odd (3). Thus, absolute values are necessary on the x.

The on the y is even (8), the of the is even (4), and the that will occur on the y once the is eliminated will be even (2). Thus, according to the rule, absolute values are not necessary on the y. (Note, though, that it would not be incorrect to have them.) The reason that the absolute values are not necessary is that y2 is nonnegative regardless of the value of y.

Thus, we have x12y84=|x|3y2. (The answer |x3|y2 is also acceptable.)

Some care is needed in applying the even-even-odd rule, as the next problem shows.


Let x be a real variable. Simplify the expression x24.

Note that, as stated, the even-even-odd rule does not apply here, since, if the were eliminated, the resulting on the x will be 12. On the other hand, it can still be used to provide a correct answer for this particular problem.



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 n is odd, the expression xn only makes sense in the real numbers when x is nonnegative.

I would like to thank Mrs. Sue Millikin, who taught me how to simplify expressions in this manner.

Title even-even-odd rule
Canonical name EvenevenoddRule
Date of creation 2013-03-22 16:01:02
Last modified on 2013-03-22 16:01:02
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Definition
Classification msc 97D40
Related topic NthRoot
Related topic SquareRoot
Related topic Radical6