Remainder

$\begin{aligned} \sqrt{2^{2048} + 3(2^{1025}) + 9} & \equiv \sqrt{(2^{1024})^2 + 2(2^{1024}\dot{}3) + 3^2} \\ & \equiv \sqrt{(2^{1024}+3) ^2} \\ & \equiv [2^{1024}+3] \pmod{6} \\ & \equiv [\color{#3D99F6}{4} + 3] \pmod{6} \quad \quad \color{#3D99F6}{\text{[See Note.]}} \\ & \equiv \boxed{1} \pmod{6} \end{aligned}$

$\color{#3D99F6}{\text{Note:}}$ From observations, we note that $\quad 2^n \mod{6} = \begin{cases} 2 & \text{when n is odd} \\ 4 & \text{when n is even} \end{cases}$