\( ( \sqrt { 8 x ^ { 6 } } ) ^ { 4 } . \)

To simplify \( ( \sqrt { 8 x ^ { 6 } } ) ^ { 4 } \), let's first break it down. The square root of \( 8 x^6 \) is \( \sqrt{8} \cdot \sqrt{x^6} = \sqrt{8} \cdot x^3 \). Since \( \sqrt{8} = 2\sqrt{2} \), we have \( (2\sqrt{2} \cdot x^3)^4 \). Raising this to the 4th power gives us \( 2^4 \cdot (\sqrt{2})^4 \cdot (x^3)^4 = 16 \cdot 2^2 \cdot x^{12} = 64x^{12} \). So, the final simplified expression is \( 64x^{12} \). Now let’s dive deeper into some interesting aspects! Did you know that square roots and exponents are interconnected concepts that originated long before modern algebra? Ancient civilizations, including the Babylonians and Egyptians, used geometric methods to solve problems similar to what we now represent with square roots. They were exploring the properties of numbers long before algebra as we know it was developed! In real-world applications, understanding exponents and square roots is essential in fields like engineering and physics. For instance, they help calculate areas, volumes, and even in algorithms for computer programming. Not to mention, they appear in models predicting population growth or radioactive decay. So next time you simplify an expression, think of all the practical uses it could have beyond the classroom!