What is the value of this complex number?

$\begin{aligned} (1+i)^8&=(1+i)^{2\times4} \\ &= (2i)^4 \\ &=\boxed{16} \end{aligned}$

We can re-write the given expression using De-Moivre's theorem: ${ \left( 1+i \right) }^{ 8 }={ \left( \sqrt { 2 } \right) }^{ 8 }{ \left( cos\left( \frac { \Pi }{ 4 } \right) +isin\left( \frac { \Pi }{ 4 } \right) \right) }^{ 8 }=16*exp\left( 2\Pi i \right) =16$