4^x - 2^(2(x-1)) +8^(2/3(x-2))>52
2^(2x) -2^(2x-2) +(2³)^(2/3(x-2))>52
2^(2x)-2^(2x-2)+2^(2x-4)>52
2^(2x) (1-2⁻² +2⁻⁴) >52
2^(2x) (1 - 1/4 + 1/16)>52
2^(2x) ((16-4+1)/16)>52
2^(2x)>52 : (13/16)
2^(2x)>52*(16/13)
2^(2x)>4*16
2^(2x)>2² * 2⁴
2^(2x)>2⁶
2x>6
x>3
х∈(3; +∞)
Ответ: х∈(3; +∞)