I am currently trying to understand the impact of lowering and raising operators on the expectation values of certain operators in a Harmonic Oscillator when the energy eigenstate has been stated as $|n\rangle$ e.g. $\hat{p}^3$
If I have a combination of lowering and raising operators I know the aim to compare their expectations value to the following
$\langle{n}|m\rangle=0$ when $m\neq{n}$
for example I have done the following
$\langle{n}|\hat{a}^3|n\rangle=\sqrt{(n)(n-1)(n-2)}\langle{n}|n-3\rangle=0$ because clearly $n\neq{n-3}$
however my difficulties are when I have combinations of lowering and raising operators, like so
$\langle{n}|\hat{a}{\hat{a}^\dagger}^2|n\rangle$
another example is
$\langle{n}|\hat{a}^2\hat{a}^\dagger|n\rangle$
I need an explanation of what $\hat{a}$, $\hat{a}^2$, $\hat{a}^\dagger$ and ${\hat{a}^\dagger}^2$ become when combined with each other in an expectation value like the ones given above.