Stickies are there for a reason.


Go to the link in my sig it is also the sticky at the top of this forum.

But to answer your question it might but its cutting it real close recommended is 650 watts. And for the 5970 I wouldn't mess with your spending $700 on that card so dont skimp on the power get atleast a 650 watt.
 
Here are the official power requirements for ATI Radeon™ HD 5970 video card.


ATI Radeon™ HD5970 System Requirements:

PCI Express® based PC is required with one X16 lane graphics slot available on the motherboard

650 Watt or greater power supply with one 75 watt, 6-pin and one 150 watt, 8-pin PCI Express® power connectors

850 Watt or greater power supply with two 75 watt 6-pin, and two 150 watt, 8-pin, PCI Express connectors for ATI CrossFireX™ technology in dual mode


The power supply recommendations are for an entire pc system.

 


Most likely it would. Corsairs are about as stout as PSU's come. They are one of the few to actually underrate their power supplies output. But it wouldn't hurt to throw another $25 at it and give yourself some breathing room.
 
Couldn't find any specs. The advertising doesn't look promising:

750W "Aywun" MEGA POWER ECO Series A1-750E Power Supply

- Peak Power 750W
- 80 PLUS
- 140mm fan
- 2 x 6+2pin and 2 x 6-pin PCIE connectors, 20+4 pin connector, 6 x IDE connectors, 6 x SATA power connectors
- Over Power Protection (OPP)
- 3-year warranty

I am always suspicious of a PSU with no specs but "peak power rating".