n^200 < 5^300. Taking the 200th root of both sides: n < 5^(300/200) = 5^(3/2) = √125 ≈ 11.18. Since n must be an integer, the largest value is n = 11. Verification: 11^200 = (11^2)^100 = 121^100, while 5^300 = (5^3)^100 = 125^100. Since 121 < 125, 11^200 < 5^300 holds true.