\begin{aligned}
 (\omega+2)^{\omega+\omega}
 & = \big((\omega+2)^\omega\big)^2 \\
 & = \Big(\sup_{n\in\omega}(\omega+2)^n\Big)^2 \\
 & = \Big(\sup_{n\in\omega}(\omega^n+\omega^{n-1}\cdot 2+\cdots+\omega\cdot 2+2)\Big)^2 \\
 & = \big(\omega^\omega\big)^2 \\
 & = \omega^{\omega\cdot 2}
\end{aligned}