# Prove rational raised to a rational is rational.

• Feb 15th 2011, 08:38 PM
jsel21
Prove rational raised to a rational is rational.
"Prove or disprove that if a and b are rational numbers, then a^b is also rational."

a = s/r, b = c/d.

(s/r)^(c/d)

(s^(c/d)) / (r^(c/d))

Now I am back where I started. Both the numerator and denominator is a rational number raised to a rational number. I'm not even sure If I am taking the right approach. Please help, thank you.
• Feb 15th 2011, 08:53 PM
pickslides
I think you have finished your proof.
• Feb 15th 2011, 09:00 PM
Prove It
Is $\displaystyle \left(\frac{1}{2}\right)^{\frac{1}{2}}$ rational?
• Feb 15th 2011, 09:11 PM
topsquark
Quote:

Originally Posted by jsel21
"Prove or disprove that if a and b are rational numbers, then a^b is also rational."

a = s/r, b = c/d.

(s/r)^(c/d)

(s^(c/d)) / (r^(c/d))

Now I am back where I started. Both the numerator and denominator is a rational number raised to a rational number. I'm not even sure If I am taking the right approach. Please help, thank you.

I think the final key here is to note that s and r aren't merely rational numbers as you say, but they are integers. And as Prove It so sagely demonstrated, integers to rational powers are not always rational.

-Dan
• Feb 15th 2011, 09:12 PM
LoblawsLawBlog
If you're not getting anywhere with a proof, it might mean that you haven't hit on the right idea, or it could mean that there can be no proof because the statement is false. See if you can find a counterexample-maybe a certain integer* raised to the 1/2 power.

*You've at least proved something: if the statement is false, then there must be a counterexample with an integer as the base.

Looks like Prove It beat me to it, so I'll just add my favorite smiley to serve as my Q.E.D symbol. (Puke)