# Thread: Properties of Roots and Absolute Values

1. ## Properties of Roots and Absolute Values

Hi. I'm in the initial review section of a pre-calc book and a little confused about something. The problem is to rationalize the denominator of the following expression:

$\displaystyle \frac{1}{\sqrt[4]{y^3}}$

The answer the book gives is

$\displaystyle \frac{\sqrt[4]{y}}{y}$

However, when I work through the problem I end up with:

$\displaystyle \frac{\sqrt[4]{y}}{|y|}$

Because of the properties of nth Roots (given in the book) which indicate that

$\displaystyle \sqrt[n]{a^n} = a\ \mbox{if n is odd}$

but

$\displaystyle \sqrt[n]{a^n} = |a|\ \mbox{if n is even}$

Have I missed something, or did the book make a mistake? I suppose it is a small point, but I keep getting confused about the issue of roots and absolute values.

2. ## Re: Properties of Roots and Absolute Values

You can usually determine the correct answer by comparing when (for instance) y = -2 and y = 2.
If y = -2, then the fourth root (of -8) is imaginary. So there must be an assumption that y is positive in this example.