Question

simplifique a expressão (2^n-1 + 2^n + 2^n+1) * (3^n-1 + 3^n + 3^n+1)/6^n + 6^n+1

Answer 1

$\dfrac{(2^{n-1}+2^n+2^{n+1})\cdot(3^{n-1}+3^n+3^{n+1})}{6^n+6^{n+1}}=\\\\\dfrac{(\dfrac{2^n}{2}+2^n+2.2^n)\cdot(\dfrac{3^n}{3}+3^n+3.3^n)}{2^n.3^n+6.2^n.3^n}=\\\\\dfrac{2^n(\dfrac{1}{2}+1+2)\cdot3^n(\dfrac{1}{3}+1+3)}{2^n.3^n(1+6)}=\\\\\dfrac{(\dfrac{1}{2}+1+2)\cdot(\dfrac{1}{3}+1+3)}{(1+6)}=\\\\\dfrac{(\dfrac{7}{2})\cdot(\dfrac{13}{3})}{7}=\\\\\dfrac{1}{2}\cdot\dfrac{13}{3}=\\\\\boxed{\dfrac{13}{6}}$