Explanation

Write (0.5)¹⁰⁰⁰ = 2⁻¹⁰⁰⁰ = 10^(−1000·log₁₀2) = 10^(−301.03). This equals 10^0.97 × 10⁻³⁰² ≈ 9.33 × 10⁻³⁰². A number of the form c × 10⁻³⁰² with 1 ≤ c < 10 has exactly 301 zeros immediately after the decimal point, so the answer is 301, option B. The common error is answering 300, the characteristic, instead of 301.