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a) Ta có:
\(2^{300}=2^{3\cdot100}=\left(2^3\right)^{100}=8^{100}\)
\(3^{200}=3^{2\cdot100}=\left(3^2\right)^{100}=9^{100}\)
Mà: \(8< 9\)
\(\Rightarrow8^{100}< 9^{100}\)
\(\Rightarrow2^{300}< 3^{200}\)
b) Ta có:
\(3^{500}=3^{5\cdot100}=\left(3^5\right)^{100}=243^{100}\)
\(7^{300}=7^{3\cdot100}=\left(7^3\right)^{100}=343^{100}\)
Mà: \(243< 343\)
\(\Rightarrow243^{100}< 343^{100}\)
\(\Rightarrow3^{500}< 7^{300}\)
c) Ta có:
\(8^5=\left(2^3\right)^5=2^{3\cdot5}=2^{15}=2\cdot2^{15}\)
\(3\cdot4^7=3\cdot\left(2^2\right)^7=3\cdot2^{2\cdot7}=3\cdot2^{14}\)
Mà: \(2< 3\)
\(\Rightarrow2\cdot2^{14}< 3\cdot2^{14}\)
\(\Rightarrow8^5< 3\cdot4^7\)
d) Ta có:
\(202^{303}=202^{3\cdot101}=\left(202^3\right)^{101}=8242408^{101}\)
\(303^{202}=303^{2\cdot101}=\left(303^2\right)^{101}=91809^{101}\)
Mà: \(8242408>91809\)
\(\Rightarrow8242408^{101}>91809^{101}\)
\(\Rightarrow202^{303}>303^{202}\)
a)\(2^{300}=8^{100}\)
\(3^{200}=9^{100}\)
\(2^{300}< 3^{200}\)
b)\(125^5=\left(25.5\right)^5=\left(5.5.5\right)^5=5^{15}\)
\(25^7=\left(5.5\right)^7=5^{14}\)
\(125^5>25^7\)
c)\(9^{20}=\left(3.3\right)^{20}=3^{40}\)
\(27^{13}=\left(3.3.3\right)^{13}=3^{39}\)
\(9^{20}>27^{13}\)
a)\(8^{300}=\left(2^3\right)^{300}=2^{900}\)
Vì \(200< 900\Rightarrow2^{200}< 8^{300}\)
b)\(25^{200}=\left(5^2\right)^{200}=5^{400}\)
Vì \(400>300\Rightarrow25^{200}>5^{300}\)
c)\(64^7=\left(4^3\right)^7=4^{21}\)
Vì \(4^{21}=4^{21}\Rightarrow4^{21}=64^7\)