Task 11.7. For what least natural $x$ is the expression

$$
\sqrt{29+\sqrt{x}}+\sqrt{29-\sqrt{x}}
$$

an integer?
Answer: 400.

Solution. If we square the given expression, we get $58+2 \sqrt{29^{2}-x}$. If the initial expression was an integer, then this number is a perfect square. We need to find the smallest $x$, i.e., we need to make this expression the largest possible perfect square. Notice that $58+2 \sqrt{29^{2}-x}<58+2 \sqrt{29^{2}}=116$. The largest square less than 116 is 100. Solving the equation $58+2 \sqrt{29^{2}-x}=100$, we get $x=400$. It is not difficult to verify that when $x=400$, the original expression is indeed an integer.