Yes you can.
Instead though, I would recommend getting (2) sticks of good quality (at least) 1866 DDR3 RAM, to first take advantage of the onboard/integrated GPU, before spending the money on a dedicated GPU, such as the HD7770.
So you want mean to get 2 gpu??? Cause i really want and apu and gpu and getting 8 gigs of ram.
No, I do not mean to get 2 GPUs.
Your APU, the A8-6600K, consists of a CPU and iGPU, which is the AMD Radeon HD 8570D. Since the APU does not have onboard/integrated VideoRAM for the integrated GPU (AMD Radeon HD 8570D), it uses your system's RAM. So, the better the RAM (the higher the frequency and the lower the latency) it has the better the integrated GPU (AMD Radeon HD 8570D) will perform. Your motherboard can only take (2) sticks of RAM, since it has only two DIMM sockets. Hence, I said, get (2) good quality sticks of RAM at a RAM frequency of at least 1866, like this: http://www.newegg.com/Product/Product.aspx?Item=N82E16820233359.
Although you can get a separate/dedicated GPU such as the HD 7770, find out how the APU runs without the HD 7770. Otherwise, you defeated the hole point of getting an APU, if you'll instant want to bypass the iGPU (AMD Radeon HD 8570D) with a dGPU like the HD7770.