Will I improve balance if I do the standing leg curl and the side hip raise without holding on?

If you have good balance it is possible to do these two exercises without holding on, but I don’t recommend it. As soon as you are not holding on, you start to concentrate more on balance than on the strengthening exercise itself. Trying to maintain balance and good form without holding on will compromise your ability to increase the amount of weight you are lifting and therefore in the end, your balance may get better but your strength won’t improve as much. If you want to do this, try gently resting your hands on the back of a chair or a counter for balance. Also, keep in mind that the exercises in which you are supporting your weight on one foot will help to improve your balance to some degree, even if you are gently supporting yourself throughout the move. If you are interested in improving your balance, I suggest you consider adding a couple of the balance exercises from “Strong Women, Strong Bones” such as the one-legged stand, tandem walk, and mountain pose. They are simple and fast, and will have a direct and positive effect on your balance without the risk of compromising form on the strength training exercises.