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Đáp án:
$243^5 = .3.27^{8}$
$3^{54} < 2^{81}$
$2^{300} < 3^{200}$
$3^{39} < 11^{21}$
$199^{20} < 2012^{15}$
Giải thích các bước giải:
•
$243^5 = (3^5)^5 = 3^{5.5} = 3^{25}$
$3.27^{8} = 3.(3^3)^8 = 3.3^{3.8} = 3.3^{24} = 3^{25}$
$=> 243^5 = 3.27^{8} = 3^{25}$
•
$3^{54} = 3^{2.27} = (3^2)^{27} = 9^{27}$
$2^{81} = 2^{3.27} = (2^3)^{27} = 8^{27}$
$=> 3^{54} > 2^{81} (9^{27} > 8^{27})$
•
$2^{300} = 2^{3.100} = (2^3)^{100} = 8^{100}$
$3^{200} = 3^{2.100} = (3^2)^{100} = 9^{100}$
$=> 2^{300} < 3^{200} (8^{100} < 9^{100})$
•
$3^{39} < 3^{40} = 3^{2.20} = (3^2)^{20} = 9^{20}$
$11^{21} > 11^{20}$
$=> 3^{39} < 9^{20} < 11^{20} < 11^{21}$
$=> 3^{39} < 11^{21}$
•
$199^{20} < 200^{20} = 200^{15}.200^5 = 200^{15}.2^5.100^5$
$2012^{15} > 2000^{15} = 200^{15}.10^{15} = 200^{15}.10^5.100^5$
$=> 199^{20} < 200^{15}.2^5.100^5 < 200^{15}.10^5.100^5 < 2012^{15}$
$=> 199^{20} < 2012^{15}$
