Question

Va rog frumos ajutatima ​

Answer 1

 

$N=\sqrt{3^{2n}\cdot5^{2n+2}+9^{n}\cdot5^{2n+3}+3^{2n+1}\cdot5^{2n+2}}\\\\N=\sqrt{3^{2n}\cdot5^{2n}\cdot5^2+(3^2)^{^n}\cdot5^{2n}\cdot5^3+3^{2n}\cdot3^{1}\cdot5^{2n}\cdot5^2}\\\\N=\sqrt{3^{2n}\cdot5^{2n}\cdot5^2+3^{2n}\cdot5^{2n}\cdot5^3+3^{2n}\cdot3^{1}\cdot5^{2n}\cdot5^2}\\\\N=\sqrt{15^{2n}\cdot5^2+15^{2n}\cdot5^3+15^{2n}\cdot3\cdot5^2}\\\\N=\sqrt{15^{2n}\Big(5^2+5^3+3\cdot5^2\Big)}\\\\N=\sqrt{15^{2n}\Big(25+125+75\Big)}\\\\N=\sqrt{\Big(15^n\Big)^2\cdot 225}\\\\N=15^n\cdot15\\\\N=15^{n+1}\in N$