5) \( \sqrt{\sqrt{8}} \)

To simplify \( \sqrt{\sqrt{8}} \), we start with the inner square root. Since \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \), we then take the outer square root: \( \sqrt{2\sqrt{2}} = \sqrt{2} \cdot \sqrt[4]{2} = 2^{3/4} \). Now, simplifying that gives us \( 2^{3/4} \), or approximately 1.68. For some math fun, did you know that nested radicals were often regarded as a sign of sophistication in historic mathematics? Ancient mathematicians like the Greeks enjoyed the elegance of such expressions, delving deep into their properties and how they could be simplified over time. If you want to explore the beauty of square roots even further, check out the history of irrational numbers! The ancient Greeks first stumbled upon them with numbers like \( \sqrt{2} \). You'll find plenty of fascinating stories about how mathematicians grappled with these esoteric concepts and how they shaped our understanding of mathematics today!