( 1/2)^x+(2/3)^x+(5/6)^x= 1
When x=3, both sides are equal.
Let f (x) = (1/2) x+(2/3) x+(5/6) x.
Because 1/2, 2/3 and 5/6 are all greater than 0 and less than 1.
So (1/2) x, (2/3) x and (5/6) x are all decreasing functions.
So f(x) is a decreasing function.
And f(3)= 1.
So x>3, f (x) <; 1
x & lt3,f(x)> 1
So this equation only holds when x=3.
So x=3.