[tex]\bf 2^{61}\;\; \boxed{} \;\; \; 3^{41}[/tex]
[tex]\bf 2^{61} = 2^{60} \cdot 2^1 =\big(2^3\big)^{20}\cdot 2^1 = (2^4)^{20}=16^{20}[/tex]
[tex]\bf 3^{41} = 3^{40} \cdot 3^1 = (3^2)^{20} \cdot 3^1 = (3^3)^{20} = 27^{20}[/tex]
[tex]\mathbf { 16^{20} < 27^{20} \iff } \color{LimeGreen}\boxed{\bf 2^{61}<3^{41} }[/tex]
[tex]\bf \;[/tex]
[tex]\bf 2^{124} \;\;\; \boxed{} \;\;\; 5^{62}[/tex]
[tex]\bf 2^{124}=(2^2)^{62} = 4^{62}[/tex]
[tex]\mathbf {4^{62} < 5^{62} \iff } \color{LimeGreen} \boxed{\bf 2^{124}<5^{62} }[/tex]
[tex]\bf \;[/tex]
[tex]\bf 2^{50} -2^{49}-2^{48} \;\;\; \boxed{} \;\;\; 3^{32}[/tex]
[tex]\bf 2^{50}-2^{49}-2^{48} = 2^{48} \Big(2^2-2-1\Big) = 2^{48}[/tex]
[tex]\bf 2^{48}=(2^3)^{16} = 8^{16}[/tex]
[tex]\bf 3^{32}=(3^2)^{16} = 9^{16}[/tex]
[tex]\mathbf {8^{16}<9^{16} \iff} \color{LimeGreen}\boxed{\bf 2^{50}-2^{49}-2^{48}<3^{32} }[/tex]
[tex]\bf \;[/tex]
[tex]\bf 5^{55} \;\;\; \boxed{} \;\;\; 55^5[/tex]
[tex]\bf 5^{55}=(5^{11})^5[/tex]
[tex]\bf 55^5=(5\cdot 11)^5[/tex]
[tex]\bf 5^3 = 125 > 55 \implies 5^{11} >55[/tex]
[tex]\mathbf{ (5^{11})^5 > (11 \cdot 5)^5 \iff } \color{LimeGreen} \boxed{\bf 5^{55}>55^5 }[/tex]
