Question

Sa se compare numerele a Si b: a=2 la puterea a 59-4la 29-8la19 Si b=3 la 39-2x9x27 la 12

Answer 1

   
[tex]a = 2^{59}-4^{29}-8^{19} = \\ = 2^{59}-(2^2)^{29}-(2^3)^{19} = \\ = 2^{59}-2^{2 \times 29}-2^{3 \times 19} = \\ = 2^{59}-2^{58}-2^{57} = 2^{57}(2^2 -2^1 -1) =2^{57}(4 -2 -1) = \boxed{2^{57}} \\ \\ b=3^{39}-2\times 9 \times 27^{12} = \\ =3^{39}-2\times 3^2 \times (3^3)^{12} = \\ =3^{39}-2\times 3^2 \times 3^{3\times12} = \\ =3^{39}-2\times 3^2 \times 3^{36} = \\ =3^{39}-2\times 3^{2+36} = \\ =3^{39}-2\times 3^{38} = 3^{38}(3-2)=\boxed{3^{38}} [/tex]


[tex]\texttt{Comparam numerele: } a = 2^{57} \texttt{ si } b= 3^{38} \texttt{ :} \\ \\ a = 2^{57} = 2^{3 \times 19}=(2^3)^{19} = 8^{19} \\ \\ b = 3^{38} = 3^{2\times 19} = (3^2)^{19}=9^{19} \\ \\ 8^{19} \ \textless \ 9^{19} \\ \\ \Longrightarrow ~~ \boxed{a \ \textless \ b} [/tex]