What is the number of zeros immediately after the decimal point in \((0.5)^{1000}\)? (Given that \(\log_{10}2 = 0.30103\))
301
\(\log_{10}(0.5)^{1000} = 1000\log_{10}(0.5) = -1000 \times 0.30103 = -301.03\). Writing \(-301.03 = -302 + 0.97\), we get \((0.5)^{1000} = 10^{0.97} \times 10^{-302}\), where \(1 < 10^{0.97} < 10\). So the number lies between \(10^{-302}\) and \(10^{-301}\), which means there are exactly 301 zeros immediately after the decimal point before the first non-zero digit.
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