Yes, you can, but it's dependant on a few things. First, you must have enough air in your lungs to blow it up. Second, there DOES come a point where the pressure from the outside water is greater than the rubber can bear, and it explodes. That's bad for you, because the air might be shover back in your lung along with water quite violently. It'd also be very hard to blow up the balloon.
________To the above...Well, yes you could. But there is no way the pressure from the outside water is going to make the balloon rupture because if the balloon is inflated, the pressure is by definition greater inside the balloon. Think of deep-sea organisms. They live under extreme pressures but doesn't need particularly sturdy bodies to cope with this. (You've probably seen those flimsy glowing jellyfish somwhere?) That's because they have a matching internal pressure that keep them from imploding. Nothing special about that, they are simply born and raised under these conditions in the same way as your body is accustomed to 1 bar of pressure. If you bring a deep sea organism to the surface quickly though, it will explode and make an awful mess.
Assuming that you would yourself be submerged to roughly the same depth as the balloon it would not be harder either, because the water would excert the same pressure on your chest and help you force air into the balloon. No matter where you are, the power needed to inflate the balloon is the same and depend on it's elastic properties. The other factors are all in equilibrium.
The one thing that differs is that you can't expect the balloon to grow as large as it would on surface level with one breath of air. If you inflate the balloon under water and bring it to the surface it will expand, yielding the same results.
A balloon can't blow itself up - it can only respond to internal and external forces. For example, if the pressure inside the balloon is greater than the pressure outside the balloon, it will inflate; if the pressure is greater than the balloon is able to stretch, it blows up. Balloons that are filled on the ground, for example, and then released into the sky may eventually blow up as they reach altitudes with lower air pressure.
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