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Răspuns:

Explicație pas cu pas:

i)

(4^3)^10 * (3^2)^10 < (5^4)^10

64^10 * 9^10 < 625^10

576^10 < 625^10 adevarat pentru ca 576 < 625

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ii)

5^66 < 3^99

(5^2)^33 < (3^3)^33

25^33 < 27^33 adevarat pentru ca 25 < 27

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iii)

2^6018 < 3^4012

(2^3)^2006 < (3^2)^2006

8^2006 < 9^2006 adevarat pentru ca 8 < 9

[tex]\it i)\ 4^{30}\cdot3^{20}=(2^2)^{30}\cdot3^{20}=2^{60}\cdot3^{20}=(2^3)^{20}\cdot3^{20}=8^{20}\cdot3^{20}=24^{20}\\ \\ 5^{40}=5^{2\cdot20}=(5^2)^{20}=25^{20}\\ \\ 24^{20}<25^{20} \Rightarrow 4^{30}\cdot3^{20}<5^{40}[/tex]

[tex]\it ii)\ 5^{66}=5^{2\cdot33}=(5^2)^{33}=25^{33}\\ \\ 3^{99}=3^{3\cdot33}=(3^3)^{33}=27^{33}\\ \\ 25^{33}<27^{33} \Rightarrow 5^{66}<3^{99}[/tex]

[tex]\it iii)\ 2^{6018}=2^{3\cdot2006} =(2^3)^{2006}=8^{2006}\\ \\ 3^{4012}=^{2\cdot2006}=(3^2)^{2006}=9^{2006}\\ \\ 8^{2006}<9^{2006}\Rightarrow 2^{6018}<3^{4012}[/tex]