Why is the domain of these two functions different? I tried graphing the functions $y = x^{.42}$ and $y = (x^{42})^{\frac{1}{100}}$. From my understanding of the laws of exponentiation these two expressions should be equivalent. What I found out after graphing these two functions was that they were equivalent for all non negative real numbers. The difference between these two functions was that the domain of $y = (x^{42})^{\frac{1}{100}}$ is $( -\infty, \infty)$ whereas the domain of $y = x^{.42}$ is $[0, \infty)$. I was wondering whether someone could explain why the domain of these two functions is different?

For $b > 0$ then $b^{n/m}=\sqrt [m]{b^n} $ is well defined whereas the same definition for $b < 0$ is not.  If $b < 0$ for example, then $b^{3/2} = \sqrt {b^3} $ is not defined as $b^3$ is negative.  But $3/2 = 6/4$ makes $b^6$ positive so a 4th root is possible.  
So note: $.42=42/100 =21/50$ but $42/100$ is not in lowest terms.  This matters as $b^{21}$ will be negative if $b $ is but $b^{42} $ will be positive. 
So for non-negative $b $, $b^{n/m} = b^{2m/2n} $ but for negative $b $ that no longer holds.
