Bài 1: So Sánh
a) Ta có: \(2^{100}=2^{10^{10}}=1024^{10}\)
\(10^{30}=10^{3\cdot10}=1000^{10}\)
mà \(1024^{10}>1000^{10}\)
nên \(2^{100}>10^3\)
b) Ta có: \(5\cdot8^{25}=5\cdot2^{75}\)
\(128^{11}=2^{77}=4\cdot2^{75}\)
mà \(5\cdot2^{75}>4\cdot2^{75}\)
nên \(5\cdot8^{25}>128^{11}\)
c) Ta có: \(8\cdot27^6=8\cdot3^{18}\)
\(9^{10}=3^{20}=9\cdot3^{18}\)
mà \(8\cdot3^{18}< 9\cdot3^{18}\)
nên \(8\cdot27^6< 9^{10}\)
d) Ta có: \(2^{100}=2^{69}\cdot2^{31}\)
\(=2^{31}\cdot2^{63}\cdot2^6\)
\(=2^{31}\cdot\left(2^9\right)^7\cdot\left(2^2\right)^3\)
\(=2^{31}\cdot512^7\cdot4^3\)
Ta có: \(10^{31}=2^{31}\cdot5^{31}\)
\(=2^{31}\cdot5^{28}\cdot5^3\)
\(=2^{31}\cdot\left(5^4\right)^7\cdot5^3\)
\(=2^{31}\cdot625^7\cdot5^3\)
Ta có: \(512^7< 625^7\)
\(4^3< 5^3\)
Do đó: \(512^7\cdot4^3< 625^7\cdot5^3\)
\(\Leftrightarrow2^{31}\cdot512^7\cdot4^3< 2^{31}\cdot625^7\cdot5^3\)
hay \(2^{100}< 10^{31}\)
