Math Olympiade Problem And Solution

Math Olympiade Problem And Solution.
Like and comment.
Share and subscribe.
Thanks 🙋🙋🙋
#maths #education #olympiad #olympiade #howtosolveolympiadmathproblem #algebraproblem #equation

[@vishalmishra3046]

*All 8 complex conjugate (including 2 real and 2 imaginary) roots of the equation*
y^x = x^y = x^(9x) = (x^9)^x. Take x'th root on both sides (since x ≠ 0 ≠ y because 0^0 is not defined) to get x^9 = y = 9x, so x^8 = 9 = 3^2 x (8 unique 8th roots of unity)^8
The only positive real root r = 3^(1/4) =~ 1.732^(1/2) =~ 1.316 and the 8 unique 8th roots of unity are e^(i 45° n) [for 0

[@palkasalyami]

Can't both x and y also be equal to 0? 🤔

[@vishalmishra3046]

No. 0^0 = x^y is not defined, so it may or may not be equal to another instance of 0^0 = y^x.

[@palkasalyami]

@@vishalmishra3046 Interesting, thanks 👍