Răspuns:
a) Exponenții 39 = 3 · 12 și 26 = 2 · 13
[tex]\bigg[\bigg(\dfrac{1}{4}\bigg)^{3}\bigg]^{13} = \bigg(\dfrac{1}{4^3}\bigg)^{13} = \bigg(\dfrac{1}{64}\bigg)^{13}[/tex]
[tex]\bigg[\bigg(\dfrac{1}{3}\bigg)^{2}\bigg]^{13} = \bigg(\dfrac{1}{3^2}\bigg)^{13} = \bigg(\dfrac{1}{9}\bigg)^{13}[/tex]
[tex]64 > 9 \Rightarrow \dfrac{1}{64} < \dfrac{1}{9} \Rightarrow \bigg(\dfrac{1}{4}\bigg)^{39} < \bigg(\dfrac{1}{3}\bigg)^{26}\\[/tex]
b) Exponenții 24 = 2 · 12 și 36 = 3 · 12
Scriem fracțiile zecimale sub formă de fracții ireductibile
[tex]0,(3) = \dfrac{3}{9} = \dfrac{1}{3}; \ \ 0,5 = \dfrac{5}{10} = \dfrac{1}{2}\\[/tex]
[tex]\bigg[\bigg(\dfrac{1}{3}\bigg)^{2}\bigg]^{12} = \bigg(\dfrac{1}{3^2}\bigg)^{12} = \bigg(\dfrac{1}{9}\bigg)^{12}[/tex]
[tex]\bigg[\bigg(\dfrac{1}{2}\bigg)^{3}\bigg]^{12} = \bigg(\dfrac{1}{2^3}\bigg)^{12} = \bigg(\dfrac{1}{8}\bigg)^{12}[/tex]
[tex]9 > 8 \Rightarrow \dfrac{1}{9} < \dfrac{1}{8} \Rightarrow \bigg(\dfrac{1}{3}\bigg)^{24} < \bigg(\dfrac{1}{2}\bigg)^{36}\\[/tex]
[tex][(0,3)]^{24} < (0,5)^{36}[/tex]
c) Exponenții 34 = 2 · 17
[tex]0,(1) = \dfrac{1}{9}; \ \ 0,(3) = \dfrac{3}{9} = \dfrac{1}{3}\\[/tex]
[tex]\bigg(\dfrac{1}{9}\bigg)^{17}[/tex]
[tex]\bigg[\bigg(\dfrac{1}{3}\bigg)^{2}\bigg]^{17} = \bigg(\dfrac{1}{3^2}\bigg)^{17} = \bigg(\dfrac{1}{9}\bigg)^{17}[/tex]
[tex]\bigg(\dfrac{1}{9}\bigg)^{17} = \bigg(\dfrac{1}{9}\bigg)^{17} \Rightarrow [(0,1)]^{17} = [(0,3)]^{34}\\[/tex]
d) 72 = 3 · 24, 48 = 2 · 24
[tex]\bigg[\bigg(\dfrac{1}{2}\bigg)^{3}\bigg]^{24} = \bigg(\dfrac{1}{2^3}\bigg)^{24} = \bigg(\dfrac{1}{8}\bigg)^{24}[/tex]
[tex]\bigg[\bigg(\dfrac{1}{3}\bigg)^{2}\bigg]^{24} = \bigg(\dfrac{1}{3^2}\bigg)^{24} = \bigg(\dfrac{1}{9}\bigg)^{24}[/tex]
[tex]8 < 9 \Rightarrow \dfrac{1}{8} > \dfrac{1}{9} \Rightarrow \bigg(\dfrac{1}{2}\bigg)^{72} > \bigg(\dfrac{1}{3}\bigg)^{48}\\[/tex]
