Question

5. Arătaţi că numărul
A=
- √(7−3√6)² + √(9–4√6)² − √(12−5√6)² + √(5-2√6)²
-
este pătratul unui număr natural.
D'C

Answer 1

$\it A=\sqrt{(7-3\sqrt6)^2}+\sqrt{(9-4\sqrt6)^2}-\sqrt{(12-5\sqrt6)^2}+\sqrt{(5-2\sqrt6)^2}=\\ \\ \\ =|\underbrace{\it7-3\sqrt6}_{ < 0}|+|\underbrace{\it9-4\sqrt6}_{ < 0}|-|\underbrace{\it12-5\sqrt6}_{ < 0}|+|\underbrace{\it5-2\sqrt6}_{ > 0}|=\ -7+3\sqrt6-9+\\ \\ \\ +4\sqrt6+12-5\sqrt6+5-2\sqrt6=-7-9+12+5=17-16=1=1^2$