\(125^5=\left(5^3\right)^5=5^{3\cdot5}=5^{15}\\ 25^7=\left(5^2\right)^7=5^{2\cdot7}=5^{14}\\ 5^{15}>5^{14}\Rightarrow125^5>25^7\)
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\(3^{54}=3^{2\cdot27}=\left(3^2\right)^{27}=9^{27}\\ 2^{81}=2^{3\cdot27}=\left(2^3\right)^{27}=8^{27}\\ 9>8\Rightarrow9^{27}>8^{27}\Rightarrow3^{54}>2^{81}\)
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\(10^{30}=10^{3\cdot10}=\left(10^3\right)^{10}=1000^{10}\\ 2^{100}=2^{10\cdot10}=\left(2^{10}\right)^{10}=1024^{10}\\ 1024>1000\Rightarrow1024^{10}>1000^{10}\Rightarrow2^{100}>10^{30}\)
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\(5^{40}=5^{4\cdot10}=\left(5^4\right)^{10}=625^{10}\\ 620< 625\Rightarrow620^{10}< 625^{10}\Rightarrow620^{10}< 5^{40}\)
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