To add $\sqrt{2}$ and $\sqrt{8}$, we begin by simplifying $\sqrt{8}$ to $2\sqrt{2}$. Then, we can proceed to add them, resulting in $3\sqrt{2}$.

When working with surds, it is essential to simplify them first. A surd refers to an irrational number that cannot be simplified to eliminate the square root. In our case, we have $\sqrt{2}$ and $\sqrt{8}$. The first step is to simplify $\sqrt{8}$.

To simplify $\sqrt{8}$, we find its prime factors. The number $8$ can be expressed as $4 \times 2$, where $4$ is a perfect square, specifically $2^2$. Thus, we can rewrite $\sqrt{8}$ as follows:

Now, we have the expressions $\sqrt{2}$ and $2\sqrt{2}$. These are like terms because they both contain the surd $\sqrt{2}$. To add them, we simply combine their coefficients—the numbers in front of the surd. The coefficient of $\sqrt{2}$ is $1$ (since $1\sqrt{2}$ is equivalent to $\sqrt{2}$), and the coefficient of $2\sqrt{2}$ is $2$.

Adding these coefficients together gives us:

Therefore, we conclude that:

This expression, $3\sqrt{2}$, represents the simplified form of the sum of $\sqrt{2}$ and $\sqrt{8}$.