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**1. Evaluate the following (i.e. get rid of the operators):**

[itex]\hat{a}^{+}\left|5\right\rangle,~~~\hat{a}\left|5\right\rangle,~~~(\hat{a}^{+})^{3}\left|2\right\rangle~~~\hat{a}^{3}\left|2\right\rangle,~~~(\hat{a}^{+}\hat{a}\hat{a}^{+}\hat{a}\left|1\right\rangle,~~~\hat{a}^{+}\hat{a}^{+}\hat{a}\hat{a}\left|1\right\rangle[/itex]

## Homework Equations

[itex]\hat{a}\left|n\right\rangle=\sqrt{n}\left|n-1\right\rangle,~~~\hat{a}^{+}\left|n\right\rangle=\sqrt{n+1}\left|(n+1)\right\rangle[/itex]

## The Attempt at a Solution

The first one is [itex]\sqrt{6}\left|(6)\right\rangle[/itex] and second one [itex]\sqrt{5}\left|(4)\right\rangle[/itex]

However I'm unsure how to evaluate for the others using the equations given. Could someone please point me in the right direction?

Thanks.