Exercise 8. Calculate

$$
\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{3+\sqrt{1}}}}}}}}}
$$

Only a numerical answer is expected here.
Solution to exercise 8 We notice the following chain of simplifications: $\sqrt{3+\sqrt{1}}=2$ then $\sqrt{7+\sqrt{3+\sqrt{1}}}=\sqrt{7+2}=3$ then $\sqrt{1+\sqrt{7+\sqrt{3+\sqrt{1}}}}=\sqrt{1+3}=2$, and so on...

Step by step, we deduce that any expression of the type $\sqrt{1+\sqrt{7+\sqrt{1+\sqrt{\cdots}}}}$ equals 2 and that any expression of the type $\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{\cdots \cdots}}}}$ equals 3, regardless of the number of square roots, provided that $4=3+\sqrt{1}$ is under the last radical.

Since a 7 is found in the sum under the first radical, the answer is therefore 3.