Find the no. of values of x belongs to R for which
64^(1/x) + 48^(1/x) = 80^(1/x)
plzz give the full soln...
39Let 1/x = t.
(64)t + (48)t = (80)t
=> 4t(16)t + 3t(16)t = 5t(16)t
=> (4t + 3t - 5t)(16)t = 0
As (16)t ≠0,
4t + 3t = 5t
We know that such a result is possible only when t = 2. You may have heard of Fermat's Last theorem, which states that for an equation of the type
xn + yn = zn, no solution exists for an integer value of n greater than 2. (x, y, z > 0)
So x = 1/2.
Otherwise, this is a Pythagorean triplet anyway.
6ya
i also got the same answer
but the answer is that there r 4 solutions!!!!!!!
how!!!
62pritish.. your statements are right but the result has been used incorrectly..
The theorem talks only about the integer solutions.. not about solutions in general...
i dont think this equation has so many solutions!
after this step you have to think of the graph of (3/5)^x+(4/5)^x.. which are both decreasing.. hence the sum is decreasing.. hence it can be equal to 1 only once..
hence there should be only one solution which can be seen as 2 from pythaorean triplets...