Being less sloppy while doing exercises I'm working through a pre-calculus course and I'm getting bad results while reviewing my exercises with the answers. It's not that I don't grasp the concepts, I make notes of the concepts, with drawings and examples I look up. If it's not clear from the book I go online and look for notes or videos (like khan academy). 
The bad results mostly boil down to being sloppy. For example, this morning I was reviewing some algebra and working through some extra exercises, like:
$7a^5b^3\cdot 5a^7b^5$. 
So I write down as an intermediate: $35a^{5+12}b^{3+5}$. Probably because I have the end result in my head also. The entire reason I want to write down intermediates is to deal with sloppiness! 
And this is a common theme throughout my work, I already parenthesise any negative numbers, since I tend to forget signs (like when doing matrices), and that really helps a lot, but errors like these are extremely common and frustrating and I don't know what to do anymore to avoid them. 'Punishing' myself by forcing myself to do more exercises doesn't seem fruitful and takes away the fun of learning.
Are there any more people here who 'suffer' from this? How do you cope with this? How do you show your work/write down your work in a structured manner? 
 A: I think of an expression as a game whose pieces are the symbols I'm manipulating. The rules of associativity, commutativity, etc let me know what the legal moves are. So when I find I'm being sloppy, I rewrite the entire expression in rows, making only one move between rows. In your example, it might look like this:
$$ \begin{align*}
& (7a^5b^3)\cdot(5a^7b^5)\\
& 7a^5b^35a^7b^5\\
& 7a^55b^3a^7b^5\\
& (7\cdot 5)a^5b^3a^7b^5\\
& (7\cdot 5)(a^5\cdot a^7)(b^3\cdot b^5)\\
& 35(a^5\cdot a^7)(b^3\cdot b^5)\\
& 35a^{5+7}(b^3\cdot b^5)\\
& 35a^{5+7}b^{3+5}\\
& 35a^{12}b^{3+5} \\
& 35a^{12}b^8
\end{align*} $$
When you force yourself to slow down and do just one manipulation at a time, you'll start to see where you're making silly mistakes and start to build good computational habits. As you get better, you'll go faster and start to do several manipulations per step - but you'll be less likely to make mistakes.
This is like playing an instrument. You can learn how to do it - but it will take work and practice. Remember that people who are able to do many steps accurately have done lots and lots of exercises already!
