that is plenty of power. you have plugged in the extra power connectors to the card right? Try re-seating the card in the slot and resetting the bios. The 6670 works so the motherboard is ok. If none of that works, return the card its probably faulty. If you did have another PC to test on that would also confirm this.
I've tried reseating it multiple times and I've also reset my bios to factory settings. It still won't boot up the computer fully. It seems to me like some sort of power issue because when I plug in the pci-e all fans turn on but nothing else gets power (my keyboard lights up if the computer can boot). Unplugging 1 pci-e lets it boot fully, but it uses the integrated gpu instead. Could the 7870 be defective in that it's got some poor power management issues or could it be that my pci-e psu cables are inefficient/broken or something?
I reset the CMOS of my MOBO by taking out the battery, still didn't work, same symptoms.
I tried putting it into the other PCI (x4 I believe) slot, the computer turned itself off after a few seconds (Same situation as the other PCI slot, except it actually turns itself off)
I tried plugging in both PCI-E PSU Power slots without plugging the GPU into the pci slot, this let the computer fully boot up normally (but then obviously no GPU).
A couple of times I reseated it back into the pci3.0 slot, the computer turned itself off just like the PCIx4 slot after I put it in there, then it went back to how it was before. (Where it turns on but doesn't boot, but without the auto turn off)