Daily Maths Challenges

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[2024/05/13] Irrational powers (lemmy.world)

submitted 4 months ago by [email protected] to c/[email protected]

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Show that it's possible a^b=c where a and b are irrational, and c is rational.

Sry for the gap I ran out of ideas.

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[–] [email protected] 1 points 4 months ago* (last edited 4 months ago)

solution

e^(i*π) = -1

Also, anything like a^(log(c) / log(a)), for positive rational c and irrational a, to generalize bean_jamming's answer

I also assert without proof that in the equation x^x = c, x is irrational for most rational values of c

I did start trying out stuff with sqrt(2), thinking back to the tower power problems, but didn't end up coming up with your solution while doing so ¯\_(ツ)_/¯