simplify(sqrt((x^2 + y^2)/x)/sqrt(x^2 + y^2)) assuming x>0, y>0

is not sqrt(1/x), why?

How should I change w1 to w2 only in exponential? For example, let

x := omega1*exp(omega1*t1*I) + omega1/exp(omega1*t1*I)

expect result:

x := omega1*exp(omega2*t1*I) + omega1/exp(omega2*t1*I)