+1072 Let a and b be positive real numbers such that a^b = b^a and b=9a. Then a can be expressed in the form
where m and n are positive integers, and n is as small as possible. Find m+n.
We can substitute b = 9a to get a^(9a) = (9a)^a. Then a = 9^(1/3), so m + n = 3 + 9 = 12.
+1072 Sorry, but your answer is wrong, maybe you made a mistake when you solved for a.
+33614 a^b = b^a. b = 9a
a^(9a) = (9a)^a
a*log(a^9) = a*log(9a)
log(a^9) = log(9a)
a^9 = 9a
a^8 = 9
a = 9^(1/8)
m = 8, n = 9
