[2024/05/13] Irrational powers
- Show that it's possible
a^b=cwhereaandbare irrational, andcis rational.
Sry for the gap I ran out of ideas.
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View original on lemmy.worlda^b=c where a and b are irrational, and c is rational.Sry for the gap I ran out of ideas.
4 replies
e^(log 2) = 2 is rational
that is simply genius
(i suppose it didnt come to me when i think of "irrational")
this one got some table slams from my friends
Hint:
::: spoiler spoiler Find an example which satisfies the equation. :::
Solution:
::: spoiler spoiler https://gmtex.siri.sh/fs/1/School/Extra/Maths/Qotd%20solutions/2024-05-13_irrational-powers.html :::
::: spoiler 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)_/¯ :::