With these types of questions it's extremely helpful to write out what you're doing and to include units. Including units is one of the best ways of making sure your answer makes sense.
Convert 63 grams of ammonia to molecules
For something like this, I would set it up as:
[itex]63 \textrm{ g }\ce{NH3}\times \frac{1 \textrm{ mol }\ce{NH3}}{17.03 \textrm{ g }\ce{NH3}}\times \frac{6.022\times10^{23} \textrm{ molecules }\ce{NH3} }{\textrm{mol }\ce{NH3}}[/itex]
Note how nicely the units cancel. After multiplying the 63 by the [itex]\frac{1}{17.03}[/itex] you'll see that there are 3.7 moles like your book says. Multiply by Avagadro's Number and you get your final answer in molecules NH3.
Glad I could help, DeadSerious!
Though I should admit that reading how I helped you kind of confuses me.
Therefore, I multiplied 6.022x10 to the 23rd power by 63(before dividing by 17.03) when I shouldn't have.
That should work. As Borek stated, the order of operations don't really matter.
(https://www.chemicalforums.com/proxy.php?request=http%3A%2F%2Fi.imgur.com%2FWTUSyry.png&hash=d2e50457d5def5f96f1b4f2f0a7128444b3c8395)
(https://www.chemicalforums.com/proxy.php?request=http%3A%2F%2Fi.imgur.com%2FYk5NQCE.png&hash=5d1ac3cb24090a8cab66e24e320f397fedb3644d)
You could multiply NA (Avagadro's number) by 63 and then divide by 17.03 or you could divide NA by 17.03 and then multiply the quotient by 63.
Regardless, I hope setting up your calculations with units helps you to see exactly what is going on. I find that it helps to make sure I'm multiplying where I'm supposed to be multiplying and dividing where I'm supposed to be dividing. And if I find that my units don't make sense then I've gone wrong somewhere.
I might find a friend/tutor/teacher who can review your approach to these questions. It sounds like you finally got it, but it also sounds like you had some trouble operating the various calculators you had. It's hard for us to see where you might be going wrong without literally watching you do it over your shoulder.