\( \sqrt { 2 ^ { 4 } } , \sqrt { ( - 3 ) ^ { 2 } } , \sqrt { 15 ^ { 2 } } \)

Let's break it down before diving into some cool insights! First, \( \sqrt { 2 ^ { 4 } } \) simplifies to \( \sqrt{16} \), which equals 4. Next, \( \sqrt { ( - 3 ) ^ { 2 } } \) simplifies to \( \sqrt{9} \), resulting in 3. Lastly, \( \sqrt { 15 ^ { 2 } } \) simplifies to \( \sqrt{225} \), giving us 15. So our final results are 4, 3, and 15! Now, here’s something fun: did you know that the concept of square roots dates all the way back to ancient civilizations? The Babylonians were already working on estimating square roots around 2000 BC! Another nugget of wisdom: when working with square roots of squared numbers, always remember they return the principal (positive) value. So, \( \sqrt{(-3)^2} \) gives you 3, not -3, which can trip up many a math enthusiast! Happy calculating!