Sometimes it's not the words itself, rather tone of your voice and overall dominance.
A nice guy with weak voice and low dominant frame: "let's go out"
>>>> even though the words suggest dominance she may not go because there is incongruance between what he says (dominant words) and his frame (weak and submissive frame)
A nice guy with weak voice and weak frame: "Let me know if you want to come"
>>>> most likely she will not go because he is not leading at all, and at the same time he is sort of asking her for permission
A dominant guy with strong frame and dominant commanding voice: "Let's go out"
>>>> she may or may not go because he is too dominant, he is pushing too much, she has no space for expressing herself, he went overboard with dominance. She may go if she is quite passive and submissive though
A dominant guy with strong frame and commanding voice: "Let me know if you want to come" or "would you like to come"
>>>> she is more likely to go because his frame is dominant and leading, yet he gave her some space to decide for herself.
Not to make it so simple, it also depends on a personality of the girl. There is a difference between self sufficient, independent and mature versus passive and inexperienced girl
If she likes you and you are congruent with your words and actions, she will go no matter how you ask her or what words you use. If she doesn't like you enough, the smartest words will not help you anyway...
So it all depends, in stead of on focusing on words see if you can focus on congruency. In general, you want to be more leading and more dominant than the girl, but you don't want to come too strong because she may fall into autorejection fast. You want to give her some choice so she can decide for herself. A good "trick" is something like: "Let me know if you want to come on Monday or Tuesday"? This way she has a space to decide for herself, however you setup a dominant frame by giving her two choices....