How many real numbers x are there such that (x(x^(1/2)))^x=x^(x*(x^(1/2))?

Question

a)1 b)2 c)3 d)4

Answer 1

Left side: x^(1/2+x)x = x^(x^2 + x/2)

Right side: x^(x^(3/2)) = x^((3/2)x)

So x^2 + (1/2)x = (3/2)x

x^2 = x

x^2 - x = 0

x(x-1) = 0

x=0 or x=1

BUT, 0 ^ 0 is not allowed

So x=1

1 is hte only answer