# 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.

Thanks in advance!

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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
Makes sense. Thanks! –  Dan M. Katz Jan 30 '12 at 1:44

## 1 Answer

Yes. $\displaystyle \left( \frac{a}{b}\right) ^{\frac{p}{q}}$ is a root of $\displaystyle b^p x^q - a^p = 0$.

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Oh, Qiaochu is so speedy. –  mixedmath Jan 30 '12 at 1:45