Let a, b, c, d be positive integers such that ` log_(a) b = 3//2 and log_(c) d = 5//4`. If (a-c) = 9, then find the value of (b-d).

Explanation

`b = a^(3//2) and d=c^(5//4)`
Let ` a= x^(2) and c=y^(4), x, y in N`
` rArr b= x^(3), d= y^(5)`
Given ` a-c=9, " then "x^(2)-y^(4) = 9`
` rArr (x- y^(2))(x+y^(2)) = 9` .
Hence, ` x- y^(2) = 1 and x + y^(2) =9`.
(No other combination in the set of positive integers will be possible.)
` x= 5 and y = 2`
`:. b - d = x^(3) - y^(5) = 125 - 32 = 93`