3 ^ ( x+3 ) - 3 ^ ( x+2 ) = 2
$\displaystyle 3^{x+ 3}= 3^x3^3= 27(3^x)$. $\displaystyle 3^{x+ 2}= 3^x3^2= 9(3^x)$.
Your equation is the same as $\displaystyle 27(3^x)- 9(3^x)= (27- 9)3^x= 18(3^x)= 2$ so $\displaystyle 3^x= \frac{2}{18}= \frac{1}{9}= 3^{-2}$.
I've always wondered about the phrase "using technology". It require technology to make a pencil doesn't it?