# 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 variable^{} is odd, then absolute value^{} 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 $n$th root (http://planetmath.org/NthRoot) of $a$ if and only if ${b}^{n}=a$ and $\mathrm{sign}(b)=\mathrm{sign}(a)$. Thus, for any positive integer $n$ and $x\in \mathbb{R}$,

$$\sqrt[n]{{x}^{n}}=\{\begin{array}{cc}|x|\hfill & \text{if}n\text{is even}\hfill \\ x\hfill & \text{if}n\text{is odd.}\hfill \end{array}$$ |

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

###### Problem.

Let $x$, and $y$ be real variables. Simplify the expression $\sqrt[\mathrm{4}]{{x}^{\mathrm{12}}\mathit{}{y}^{\mathrm{8}}}$.

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 ${y}^{2}$ is nonnegative regardless of the value of $y$.

Thus, we have $\sqrt[4]{{x}^{12}{y}^{8}}={|x|}^{3}{y}^{2}$. (The answer $|{x}^{3}|{y}^{2}$ is also acceptable.)

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

###### Problem.

Let $x$ be a real variable. Simplify the expression $\sqrt[\mathrm{4}]{{x}^{\mathrm{2}}}$.

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 $\frac{1}{2}$. On the other hand, it can still be used to provide a correct answer for this particular problem.

Solution:

$$\sqrt[4]{{x}^{2}}=\sqrt{\sqrt{{x}^{2}}}=\sqrt{|x|}$$ |

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 $n\in \mathbb{N}$ is odd, the expression $\sqrt{{x}^{n}}$ 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 |