# Is every rational number raised to a rational power an algebraic number?

Question is pretty much explained in the title. My inclination is to say "yes", but I'm unsure.

Sure; pretty much by definition, $r^{p/q}$ is a root of the polynomial $x^q = r^p$. – Qiaochu Yuan Jan 30 '12 at 1:42
Yes. $\displaystyle \left( \frac{a}{b}\right) ^{\frac{p}{q}}$ is a root of $\displaystyle b^p x^q - a^p = 0$.