I'm deriving the action of the squeezing operator on a non-vacuum Fock state and I have almost finished but can't work out how to apply an operator that is stuck in the power of a constant, not an exponential. I am trying to work out the result of this action:

$$\bigg(\frac{1}{\cosh{|\zeta }|}\bigg)^{\hat{a^{\dagger}} \hat{a} +\hat{b^{\dagger}}\hat{b} + 1} |{0,b}\rangle $$

I suspect the solution is:

$$\frac{1}{\cosh{|\zeta }|}\sqrt{b}|0,b\rangle,$$

but I'm not certain how to show this. I know you can Taylor expand and exponentiated operator but that doesn't apply here.