5. Determine whether the number $\sqrt[2023]{3+\sqrt{8}}+\sqrt[2023]{3-\sqrt{8}}$ is greater or less than two.

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# Solution

The given addends are reciprocals of each other, since

$$
\frac{1}{3+\sqrt{8}}=3-\sqrt{8}
$$

It is clear that they are not equal to each other.

Let $a=\sqrt[2023]{3+\sqrt{8}}$. Then the given expression is $a+\frac{1}{a}$, and the sum of two different positive reciprocals is strictly greater than two.

Answer: $\sqrt[2023]{3+\sqrt{8}}+\sqrt[2023]{3-\sqrt{8}}>2$