Question

calculeaza 3^3^3^3 : 3^3^5^2 = 3^27 : 3^30

Answer 1

$\\ Calculam cele doua numere separat.$

$\boxed{1} \quad Nu avem paranteze, calculam puterile de sus in jos: \\ \\ {{3^3}^3}^3 : {{3^3}^5}^2 = {3^3}^{27} :{3^3}^{25} = (3)^{\big3^{\big{27}}}:(3)^{\big3^{\big{25}}} = 3^{\big{3^{27}-3^{25}}} = \\ \\= 3^{\big{3^{25}(3^2-1)}} =3^{\big{3^{25}(9-1)}} = 3^{\big{3^{25}\cdot 8}} \\ \\ \\ \boxed{2} \quad 3^{27}:3^{30}} = 3^{27-30} = 3^{-3}$

$\\ Comparam cele 2 numere: \left\| \begin{array}{c}3^{\big{3^{25}\cdot 8}} \boxed{?} 3^{-3}\\ \\ -baze supraunitare egale. \\ -exponenti diferiti:\\ {\big{3^{25}\cdot 8} \ \textgreater \ -3 \end{array} \right |\Rightarrow 3^{\big{3^{25}\cdot 8}} \ \textgreater \ 3^{-3} \\ \\ \\ \Rightarrow \boxed{\boxed{{{3^3}^3}^3 : {{3^3}^5}^2 \ \textgreater \ 3^{27}:3^{25} }}$

$\\ In concluzie, cele doua numere nu sunt nicidecum egale.$