Question

Daca 36 la puterea a =4 si 36 la puterea b = 3 sa se arate ca 25 la puterea 1-a-b supra 1-a apartine Z

Answer 1

[tex]36^b=3 \ \ \ (1)\\ 36^a=4\ \ \ \ (2)\\\;\\ 4=\dfrac{36}{9}=\dfrac{36}{3^2}\stackrel{(1)}{=}\dfrac{36}{36^{2b}} \ \ \ (3)\\\;\\ (2),\ (3) \Longrightarrow 36^a=\dfrac{36}{36^{2b}} \Longrightarrow 36^a\cdot 36^{2b} =36\Longrightarrow 36^{a+2b} =36 \\\;\\ \Longrightarrow a+2b=1 \Longrightarrow 2b=1-a\ \ \ (4) [/tex]

Evaluam exponentul lui 25 :

$\dfrac{1-a-b}{1-a}\ \stackrel{(4)}{=} \ \dfrac{2b-b}{2b}=\dfrac{b}{2b}=\dfrac{1}{2}$

Deci, vom avea :

$25^{\frac{1}{2}} =\sqrt{25} =5 \in \mathbb{N}$