[tex]\it a=\dfrac{2^n}{3^n}:\dfrac{2\cdot2^n+6\cdot2^n\cdot3^n}{3\cdot3^n+9\cdot3^{2n}}=\dfrac{2^n}{3^n}\cdot\dfrac{3^n(3+9\cdot3^n)}{2^n(2+6\cdot3^n)}=\dfrac{3+9\cdot 3^n}{2+6\cdot3^n}=\\ \\ \\ =\dfrac{3(1+3^{n+1})}{2(1+3^{n+1})}=\dfrac{3}{2}[/tex]
[tex]\it \Big(\dfrac{2a}{3}\Big)^{100}=\Big(\dfrac{2}{3}\cdot a\Big)^{100}=\Big(\dfrac{2}{3}\cdot \dfrac{3}{2}\Big)^{100}=1^{100}=1[/tex]
