Explanation:

To compare these large numbers, the most effective method is to express them with a common exponent. The exponents are $500, 400, 300, 200$. The greatest common divisor (GCD) of these exponents is $100$.\nWe can rewrite each number in the form $(X^k)^{100}$. \nLet's evaluate each option: 1. $2^{500}$: This can be written as $(2^5)^{100}$. $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. So, $2^{500} = 32^{100}$. 2. $3^{400}$: This can be written as $(3^4)^{100}$. $3^4 = 3 \times 3 \times 3 \times 3 = 81$. So, $3^{400} = 81^{100}$. 3. $4^{300}$: This can be written as $(4^3)^{100}$. $4^3 = 4 \times 4 \times 4 = 64$. So, $4^{300} = 64^{100}$. 4. $5^{200}$: This can be written as $(5^2)^{100}$. $5^2 = 5 \times 5 = 25$. So, $5^{200} = 25^{100}$. \nNow we have all numbers expressed with the same exponent, $100$: $32^{100}, 81^{100}, 64^{100}, 25^{100}$.\nWhen comparing numbers with the same positive exponent, the number with the largest base is the largest.\nComparing the bases: $32, 81, 64, 25$. The largest base is $81$.\nTherefore, $81^{100}$ is the largest number, which corresponds to the original expression $3^{400}$. \nLet's evaluate the options:\nOption (1) $2^{500} = 32^{100}$: Smaller than $81^{100}$.\nOption (2) $3^{400} = 81^{100}$: This is the largest value.\nOption (3) $4^{300} = 64^{100}$: Smaller than $81^{100}$.\nOption (4) $5^{200} = 25^{100}$: Smaller than $81^{100}$. \nThus, the largest number is $3^{400}$. The correct option is (2) $3^{400}$.