quantum mechanics - Position and momentum eigenstates in terms of creation and annihilation operator?

Consider a simple harmonic oscillator; the position operator is $\hat{x}=(a^\dagger+a)/\sqrt{2}$ and the momentum operator is $\hat{p}=-i(a-a^\dagger)/\sqrt{2}$.

One may verify that the eigenstates of $\hat{x}$ and $\hat{p}$ are $$\left|x\right>\propto e^{\sqrt{2}~xa^\dagger-(a^\dagger)^2/2}\left|0\right>$$ $$\left|p\right>\propto e^{ip\sqrt{2} ~a^\dagger+(a^\dagger)^2/2}\left|0\right>.$$ My question is: how do I verify that the position eigenstates and momentum eigenstates are orthogonal themselves, and that $$\left\propto e^{ipx} ~~?$$ I'm not able to calculate this inner product using the commutator of $a$ and $a^\dagger$.