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Question

If a and b are positive numbers such that a^b = b^a and b = 9a, then the value of a is

Accepted Answer

Substituting b = 9a into a^b = b^a gives align*a^9a = (9a)^a & (a^9)^a = (9a)^a (using the identity (x^y)^z = x^yz for x > 0) & a^9 = 9a (taking the ath root of both sides, as a > 0) & a^8 = 9 (as a 0) & a^4 = 3 (taking square roots and noting that a^4 0) & a = [4]3 (again as a > 0).align*