= Solution
Invert the given <ladder operators>:
$$
\hat x=\sqrt{\frac\hbar{2m\omega}}(\hat a_++\hat a_-),\qquad\hat p=i\sqrt{\frac{m\hbar\omega}2}(\hat a_+-\hat a_-).
$$
Because the <Energy eigenstate> are orthonormal, $\langle n|\hat a_\pm|n\rangle=0$: each operator changes the state to one orthogonal to $|n\rangle$. Hence
$$
\boxed{\langle x\rangle_n=\langle p\rangle_n=0.}
$$
The result also follows from the definite <parity> of every oscillator energy eigenfunction, but here it follows directly from the specified operators.
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