## Solution to Exercise 1 We have:

$\sqrt{32 \sqrt{16 \sqrt{8 \sqrt{4}}}}=\sqrt{32 \sqrt{16 \sqrt{8 \cdot 2}}}=\sqrt{32 \sqrt{16 \sqrt{16}}}=\sqrt{32 \sqrt{16 \cdot 4}}=\sqrt{32 \sqrt{64}}=\sqrt{32 \cdot 8}=\sqrt{256}=16$.

Alternative Solution $n^{\circ} 1$ Using the multiplicativity of the square root, we find:

$$\begin{aligned} \sqrt{32 \sqrt{16 \sqrt{8 \sqrt{4}}}} & =\sqrt{32} \cdot \sqrt[4]{16 \sqrt{8 \sqrt{4}}} \\ & =4 \sqrt{2} \cdot \sqrt[4]{16} \cdot \sqrt[8]{8 \sqrt{4}} \\ & =4 \sqrt{2} \cdot 2 \cdot \sqrt[8]{8} \cdot \sqrt[16]{4} \\ & =8 \cdot 2^{1 / 2} \cdot 2^{3 / 8} \cdot 2^{2 / 16} \\ & =8 \cdot 2^{1 / 2+3 / 8+1 / 8} \\ & =8 \times 2=16 \end{aligned}$$

Grader's Comment: The exercise is very well done. The few errors are due to poor manipulation of the square root.